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ACTUARIAL STUDIES NO. 2

CONSTRUCTION OF MORTALITY TABLES
FROM THE RECORDS OF INSURED LIVES

t ^^RINCIPAL CONTRIBUTORS j.

RAY D. MURPHY AND PERCY C. Hf PAPPS

PUBLISHED BY

THE ACTUARIAL SOCIETY OF AMERICA

256 BROADWAY, NEW YORK

1922

ORIGINAL COMMITTEE
Arthur Hunter Wendell M. Strong

Henry Moir A. A. Welch

P. C. H. Papps a. B. Wood

John K. Gore, Chairman

COMMITTEE NOW IN CHARGE

Robert Henderson Wendell M. Strong

J. M. Laird J. S. Thompson

A. T. Maclean Hugh H. Wolfenden

A. H. Mowbray Henry Moir, Chairman

Thb Actuarial Societt of America

New York

PRESS OF

THC NEW EIM PRINTINQ COMPANY

LANCASTER, PA.

CONSTRUCTION OF MORTALITY TABLES FROM THE
RECORDS OF INSURED LIVES.

Chapter I. Introduction.

A mortality table starts with a group of persons at a specified
age and shows the number of survivors at each subsequent age.
There is also generally set down the number dying in each year
of age. The radix of the table, or the number living at the
youngest age shown, is arbitrarily selected. Then, from the
values of px for each age, the values of h may be computed
successively for the higher ages by the relation, Upx = h+i*

As a general rule the construction of a mortality table based
on the records of insured lives is effected by ascertaining the
value of 5x for each a^e. This vahie is obtained by dividing
6x by Ex, where 6x represents the deaths and Ex the exposed to
risk of death for the year between ages x and x + 1 in the data
under observation.!

In practice some persons will be found who are under observa-
tion for only a part of a given year of age either because they
enter the experience after the beginning of the year or because
they pass out of it for causes other than death before the end
of the year. In such cases each person is counted in computing
Ex as a fraction equal to the proportion of the year under ob-
servation. The degree of accuracy with which such fractions are
computed varies according to the method used in tabulating the
data. A person who dies between ages x and x + 1 must be
included in Ex as exposed to risk for the full year and not a
fraction, because Qx represents the proportion of Ix persons alive
at age x who will not be alive at age x -\- I.

* See Institute of Actuaries' Text Book, Part II, Chapter I.

t As will be seen in Chapter III, 6x may be the number of deaths and Eg
the number of Uves exposed to risk of death, or they may represent respectively
the mmiber of policies terminated by death and the number exposed to risk
of termination by death; or lastly they may be the amoimt of insurance
terminated by death and the amount exposed to risk. It is important to
note, however, that both numerator and denominator in the fraction dx/E,
must always relate to the same kind of data.

2 CONSTRUCTION OF MORTALITY TABLES

It should be remembered that theoretically the numerical
value of the radix selected and the resulting size of the values of
Ix and dx are of no significance, but that the relative values of
these functions are of vital importance. The use of a large radix
is advisable, however, so that when the calculated values of
Ix and dx are adjusted to the nearest integer the necessary error
introduced is insignificant. This will also have a bearing on the
limiting age (w) since this would be the lowest age for which the
unadjusted value of h is less than .5, unless fractions are shown.

When an applicant is accepted for insurance after being ex-
amined by the company's physicians, he is a "select" life.
Among a number of such lives after the lapse of a few years there
will be some whose health has become impaired to a greater or
less degree, while others will remain as healthy as when first
examined for insurance. The survivors of a body of select lives
are therefore called "mixed" lives.

It follows that the rate of mortality of insured persons of a
given attained age, say x, will not be the same among persons
just insured at age x as among persons insured n years ago at
age X — n. It has been found by experience that persons just
insured at age x are subject to a lower rate of mortality than
those also aged x, but insured at age x — 1. These latter, in
turn, usually show a lower rate of mortality than those aged x,
but insured at age x — 2. In general, for limited values of n,
which vary in different experiences and in different age sections
of a single experience, it is found that g[x_^in')+;nT < 9[x-ni+n
where the portion of the suffix within the square brackets in-
dicates the age at issue, and the other portion, the duration since
entry, the total being the present age, i.e., x.

As n increases, the extent of the difference will be found to
decrease, so that if g[i_;izi]+^:ii = ?[*-n]fn — 5„ then as n in-
creases, 5„ will approach the limit zero.

If t be the greatest value of n for which the relation ^[x-iiZii+s:!^^
< 9[i_„]+n holds, this fact is expressed by saying that the effects
of selection last for t years. Accidental fluctuations in the data,
on which the mortality rates are based, are alone sufficient to
prevent any exact determination of the value of t. An approxi-
mate value is all that can be expected. In practice, t may have
a small value, as, for instance, in the case of residents in the
tropics, where values as low as 2 years or even 1 year may be
found; or it may have such a comparatively large value as 10

FROM THE RECORDS OF INSURED LIVES. 6

years or more, the latter figure applying in the case of the British
Offices' Experience (1863-1893) under whole life participating
policies, especially at the younger ages. It may be questioned
whether the effects of selection ever entirely disappear or whether
they become so merged with other influences, such as changes in
sanitary conditions and in the mortality of the general popula-
tion, that they are lost.* As a practical matter, however, we
are warranted in assuming that they cease after a certain period.

Now, if a body of select lives all of a given age be observed,
and the rate of mortality resulting during the first, second,
third, etc., years of insurance be set forth for each year, the
result will be a select table of mortality for that particular age
at entry. If similar tables be prepared for each age at date of
selection, we obtain a set of "select mortality tables."

It would involve much labor, however, to base calculations on
such a set of select tables. A trial is therefore made to ascertain
the effective period of selection beyond which the rate of mor-
tality appears to depend only upon the attained age and may
consequently be formed into one "ultimate" table. This may
be done by a direct comparison of the values found for qix-n]+n
for different values of n as indicated above. Such a comparison
may be confusing, however, because of the large number of
values to be observed and the fluctuations in them, and it will
usually be more satisfactory to determine by observation ap-
proximately where the line of division lies and then apply a
different final test.

In describing the construction of the American Men Tables, it
was stated that "the crude death rates were deduced for each
of the first five insurance years, for the sixth and succeeding
years combined, and for the eleventh and succeeding years
combined. The expected deaths for each of the first ten in-
surance years were then calculated by graded rates of mortality
based upon the data for the sixth and succeeding insurance years
in order that the number of years for which medical selection
lasted could be determined. It was seen that the material for
the sixth and succeeding insurance years could be safely com-
bined according to attained age."

A different method was used in compiling the O^^^ table. The
expectations of life were employed, as they would not be subject

* See T.A.S. A., Vol. XIII, page 211, for a discussion of the effect on select
tables of a variation in mortality during the period of investigation.

CONSTRUCTION OF MORTALITY TABLES

to fluctuations to the same degree as would the mortality rates
for individual ages. On page 146 of "Account of Principles and
Methods" of that experience are shown, for quinquennial groups
of ages, the values of the expectations eix], e[x_5]+5, e[i_io]+io, etc.,
to e[x_2B]+75. There are also given the values of ej-^^ and ex^^*^^}
the expectations of life found by combining the data for the same
attained age, but excluding the data for the first five and first
ten years of duration respectively. If selection were still effect-
tive in the (n + l)th year, e[i_„]+„ would be greater than Cx^"^.
It was decided for practical advantages to consider that the effect
of selection had disappeared in ten years, although this did not
appear to be true for all ages at entry.

A select and ultimate table may be set forth conveniently as
shown by the following section of the O"^^'^^ table.

Age at
Entry.

Years Elapsed Since Date of Insurance.

Age At-
tained.

. 0

6 or
more.

Im-

hx]+i-

hx]+t-

l[x]+i.

^(x]+4.

llx]+t.

1+5.

20
21
22
23
24
25
26
27
28
29
30
31
32
33

100,000
99,264
98,530
97,794
97,055
96,316
95,567
94,818
94,059
93,300
92,529

99,580
98,844
98,109
97,369
96,630
95,887
95,135
94,382
93,618
92,854

99,003
98,267
97,530
96,790
96,048
95,302
94,547
93,791
93,023

98,333
97,596
96,857
96,115
95,370
94,619
93,862
93,100

97,616
96,877
96,135
95,389
94,639
93,884
93,122

96,879
96,137
95,392
94,641
93,886
93,124

25
26
27
28
29
30
31
32
33
34
35
36
37
38

When the rates of mortality are obtained, the first line of the
table may be started with the desired radix and the values suc-
cessively computed across the first line and then down the last
column. The second and subsequent lines may be calculated
by working back from the ultimate column by means of the
equality log lix]+n-i = log lix]+n — log p^+n-i-

If the data entering into both the select and ultimate sections
of the table be combined, or, in other words, if a mortality table
be formed according to age only, irrespective of the year of
insurance, the result will be an "aggregate table."

FROM THE RECORDS OF INSURED LIVES. 5

While it may be known that the effects of selection last for
several years, it may be thought desirable for practical purposes
to construct a table of mortality excluding the experience of,
say, the first two years only, without constructing the select
tables corresponding to those first two years. Such a table is
known as a "truncated" table. Every ultimate table is in a
sense a truncated table, but the name "ultimate" is usually
applied only to a table which forms the continuation of the
select section of a mortality table. In the select section the
rate of mortality is shown for the age attained, but modified
according to the length of time elapsed after initial selection.
In an aggregate or an ultimate table the rate of mortality is
shown for each age attained, without modification.

It is desirable to consider the effects of the duration of in-
surance of the data entering into an aggregate table. It will
be understood that the following remarks will apply generally
but in a modified degree to a truncated table. By first obtaining
a clear idea of the nature of these different forms of mortality
tables, the student will be in a better position to grasp the sig-
nificance of the various methods of investigating and collecting
the data.

For the sake of illustration, let it be first assumed that at all
ages the effects of selection will last for ten years only. Then,
if an aggregate table be formed from the experience of a company
that has been in business for only ten years, the resulting table
will be composed only of lives which have not reached the ultimate
rates of mortality. The rates of mortality shown will obviously
be much less than will be the case when the table is based upon
the total experience of an old company which has been many
years in business; for in the latter case, the higher ultimate
rates of mortality of the old business will be included, raising
the aggregate rates for any given age above the lower mortality
of the newer business for the same age. Again, if there be two
companies of the same age, the aggregate tables formed from the
experience of the respective companies will differ considerably
if one company has recently been writing a much larger business
in relation to its size than the other. The company having the
larger proportion of select business in its aggregate data will,
other conditions being the same, show the lower mortality ex-
perience. In making this statement it is assumed that the age
distribution is similar and that the companies are subject to

6 CONSTRUCTION OF MORTALITY TABLES.

the same select rates of mortality, the difference in the aggregate
mortality being due solely to the different proportions of new and
old business.

From what has been said it will be clear that a select table is
the true measure of the adequacy of premiums for life insurance.
The great convenience of an aggregate or an ultimate table for
actuarial calculations however makes the use of such a table
desirable if it can be shown to produce adequate results. The
Combined Experience, the H^ and O^ Tables are aggregate tables,
the American Experience an ultimate, and O^^*^ a truncated
aggregate table. These tables have served a very useful purpose
as the basis of insurance premiums and reserves.*

For a basis of comparison in mortality investigations select
tables are essential to prevent erroneous conclusions due to the
difference in the average policy duration of two or more classes
of policyholders.

* See T. A. S. A., Vol. XII, page 49, for application of the principle of
truncated tables.

Chapter II . Sources of Data.

Any records showing the distribution according to ages or
groups of ages of persons living and the number of deaths oc-
curring among such persons, may be used for the construction
of a mortality table, provided there are a sufficient number under
observation to permit the law of average to operate and to show
the general trend of the mortality. The larger the number of
lives involved, the more reliable, as a general rule, will be the
resulting mortality table. While it is preferable to be able to
ascertain the exposed to risk and the deaths for each age, it is
not essential; for if the average values of qx for groups of ages are
known, a mortality table may be constructed by interpolation
showing the values of qx for every age.*

Population statistics have been utilized for compiling mor-
tality tables. With the properly compiled records of two censuses
and of the intervening deaths a reliable table may be constructed.
The Carlisle Table and the English Life Tables are examples of
tables which on the whole are properly constructed, while the
Northampton Table is an example of an unsuccessful attempt to
construct a mortality table from the records of deaths only.
The construction of a mortality table from population statistics
is covered in No. 3 of Actuarial Studies, and the characteristics
of the tables mentioned are set forth in No. 1.

The records of insurance companies and fraternal societies
form the most valuable source for compiling mortality tables
for the use of such companies and societies. The tables are
required for measuring the mortality among practically the same
class of lives as that on which the tables are based. The Amer-
ican Experience Table may be cited as one based on the experi-
ence of an American life insurance company, namely. The
Mutual Life Insurance Company of New York. The American
Men Table was formed from the combined data of many Amer-
ican and Canadian companies showing the experience on Amer-
ican males. The Combined Experience, or, as it is sometimes
called, the Actuaries', or Seventeen Offices' Table, the H^ Table
and the 0^ Table are based on the experience of groups of

* See Actuarial Studies No. 4.

8 CONSTRUCTION OF MORTALITY TABLES.

British companies. The experience of a number of fraternal
societies was used as the basis of the National Fraternal Congress
Table. These tables are described in No. 1 of Actuarial Studies.
Among the various miscellaneous sources from which mortality
tables may be compiled are the volumes, published annually,
giving details concerning the families of the British Peerage.
These volumes supply data relating to British Peers, the sons
and daughters of Peers and the sons and daughters of the eldest
sons of Peers, from which a number of Peerage Tables of mor-
tality have been constructed. The records kept by universities
have also been employed in compiling a mortality table. From
the records of Widows' and Pension Funds tables of mortality
may be formed which will be useful as a guide to the management
of such funds.

Chapter III. Methods of Recording the Data.

As this study deals only with mortality tables from the
records of insured lives the methods hereafter described will be
those applicable to the data available in an insurance company
or fraternal societJ^ The facts regarding each policy will be
set down on a card so that they may be sorted and tabulated
(See Chapter IV). The details to be recorded will depend partly
upon the use for which the mortality table is desired, and methods
may be varied to facilitate the work, provided accuracy is not
thereby sacrificed.

There are three possible bases for determining 5^, (1) the death
rates among the lives insured, (2) the rates of termination by
death of policies in force, and (3) the rates of termination by
death of the amounts insured.

If it is desired to investigate by lives, the data concerning all
the policies on each life must be brought together to avoid
duplication. Where the experience is confined to one company
the office records will usually show all policies on a single life,
but when the combined experience of several offices is under
investigation, the existence of duplicate policies in different
offices can be ascertained only by examination. The usual plan
is to arrange all names in strict alphabetical order — as in a
directory — and thus enable the bulk of the duplicates to be
brought together. Where names agree, but the dates of birth
differ radically, different lives may be presumed ; where the dates
of birth differ slightly, it may sometimes be found that the cards
refer to a single life. Occasionally day and month of birth will
agree, but not the year. Inquiry as to occupation and residence
will usually throw light upon doubtful cases.

Some persons, however, are careless in the use of their names,
and it will occasionally happen that a name may appear as, say,
George Frederick Smith in the records of one company, and
Frederick Smith in those of another — or the surname may be
written Smyth. Alphabetical sorting will fail to disclose the
existence of such duplicates, and, where complete elimination is
considered important, an independent sorting of the material
by dates of birth should be effected. This latter device is

10 CONSTRUCTION OF MORTALITY TABLES

especially useful in the case of female lives where change of name
owing to marriage is quite often recorded in one office and not
in another, owing to carelessness on the part of the insured, or
to the termination of a policy prior to marriage. Even so,
errors in dates of birth will prevent complete elimination of
duplicates.

Having brought the duplicates together the method of elimina-
tion depends upon the kind of table that is to be constructed.
If it is an aggregate table and there has been a continuous ex-
posure from the date of entry of the earliest policy to the last
date of exit, one card may be written for the complete exposure
and the others destroyed. If there has not been a continuous
exposure then two or more cards must be written to cover the
several continuous periods that the company was at risk. If
select tables are to be compiled the life should count only once
in the ultimate section of the table, but may be counted once for
each select period beginning at a different age. This is seen from
the fact that each such select period is a separate experience and
it involves no duplication to allow one life to enter two or more
such experiences.

A difficulty enters here because the duration of the select
period cannot be determined in advance. One card may there-
fore be written for the complete exposure and separate cards for
the other select periods, using an assumed duration for the period
of selection. These latter cards should be coded so that they
may subsequently be sorted out and the durations corrected
after the select period has been decided upon.

The British Offices' Experience illustrates the practical diffi-
culties which may arise in eliminating duplicates. Several
sections of the experience were to be compiled independently,
combining the data in some of these afterward. Furthermore
aggregate, truncated and select tables had to be provided for.
Strict elimination in every case would have been too laborious
and consequently duplication was considered in each section
only. When sections were combined some lives were therefore
recorded more than once. Furthermore one card could not
well be written for the continuous exposure on one life for the
aggregate table, as this would conceal the select experience
arising from policies issued after the earliest one which should
properly enter the select tables, and also would make the trun-
cated tabulation desired impossible. A card was consequently

FROM THE RECORDS OF INSURED LIVES. H

written for each age at which a policy was issued but such cards
were marked to enter the aggregate experience only at the dura-
tion corresponding to the date of exit of the preceding policy.
The result was that the select tables contained duplication in the
ultimate section arising from policies on one life issued at dif-
ferent ages. The truncated table likewise contained two or
more coincident exposures on some lives. The investigation was
therefore not based wholly on lives but on a combination of
lives and policies.*

Mr. G. F. Hardy has stated, in "The Theory of the Construc-
tion of Tables of Mortality, etc.," p. 18:

"Without dogmatizing upon the point, it appears to me that the proper
course is, where two or more policies are effected at the same time or at the
same age at entry, to treat them as a single risk, but where the subsequent
poUcies are effected at later ages, involving fresh medical selection, to treat
them as separate risks. This means the elimination of duphcates in each of
the 'select' tables for individual ages at entry, but no fm-ther elimination in
the resulting aggregate tables, a course which has the advantage of making the
aggregate table the true aggregate of the tables for separate ages at entry.
Judging by the results of the 0" experience, this course is necessary if we are to
produce an aggregate table, representing 'ultimate' rates of mortality after
the lapse of a stated period from entry, which will join on smoothly to the
'select' rates."

If the investigation is by policies there will be cases where one
life is insured under several policies and the failure of one such
life will have a greater effect upon the mortality table than the
failure of a life insured under a single policy. Now, if the lives
which are insured under several policies are on the whole subject
to a lower rate of mortality than lives of similar ages insured
under single policies, it follows that the table based on policies
will show lower rates of mortality than if the table had been
based on lives and vice versa. The general effect of investigating
by policies is to produce rates of mortality fairly close to those
determined by lives insured except for accidental fluctuations
where the exposures are small; but it has been thought that
an experience by policies may show a slight tendency to diminish
the values of q^ at the younger and increase them at the older
ages in an aggregate table. This is probably due to the fact
that at the younger ages, where some individuals are repeatedly

* For an instructive detailed account of the manner in which the cards were
marked and duphcates ehminated see the volume of the British OflGices' Life
Tables, 1893, entitled "Account of Principles and Methods."

12 CONSTRUCTION OF MORTALITY TABLES

undergoing medical examination to obtain additional policies,
those who are successful represent a class of superior lives, and
the additional weight given to them in the experience tends to
lower the mortality; while those who are unsuccessful do not
show a proportionate effect. At the older ages where few addi-
tional policies are taken this extra weight operates to increase the
proportion that the ultimate data bears to the whole.

The saving of labor in investigating by policies rather than by
lives will frequently justify the former method. The result will
usually be a satisfactory basis for comparison with other mor-
tality tables and with an insurance company's experience in
special classes of risks. Neither of these methods however is
necessarily a safe basis for the computation of premiums and
reserves, since the rate of financial loss may be greater than the
rate of mortality by lives or policies. This will be the case
when the mortality among lives insured for large amounts is
greater than among those insured for small and moderate
amounts.

Investigations by amounts insured, in which qx is determined
as the ratio of claims incurred to insurance in force, are con-
sidered essential under such circumstances, which are common
in this country. It is evident that mortality rates so derived
will be subject to accidental fluctuations caused by the failure
of one or more lives insured for large amounts. Such fluctua-
tions may subsequently be removed by graduation or may be
lessened by restricting the limit of insurance for which any one
life will be counted as exposed in the experience. This latter
restriction would necessitate the labor of bringing together for
investigation all policies on any life insured for a large amount.
The use of amounts insured introduces additional work also
because of the necessity for noting all changes in amount after
issue.

In the American-Canadian Mortality Investigation (1900-
1915) it was decided to count as $100,000 only, insurance on any
individual life issued at any one age for more than that sum. In
order to accomplish this purpose there was noted on the cards
furnished by each company the insured's initials and full date of
birth if the policy was for $50,000 or more. For each age at
entry those cards were brought together which had the same
date of birth and initials. The mortality was to be investigated
also by plans of insurance and Volume I of that experience

FROM THE RECORDS OF INSURED LIVES. 13

states, "When more than $100,000 of insurance was issued at
the same entry age on the same life but on different plans of
insurance, the amount of insurance on each card was reduced
proportionately, provided the mode of termination and duration
were identical. For instance, if $50,000 had been issued on the
twenty-payment life plan and $100,000 on the ordinary life
plan, these amounts were reduced to $33,300 and $66,700 re-
spectively. This procedure made it unnecessary to make any
further adjustment when the investigation was made by plan
of insurance.

"When the duration of the several policies differed, the
policy with the longest duration was retained in the investigation
for its original amount, provided such amount did not exceed
$100,000."

This method guarded against an undue effect during the
select period of the failure of a life insured for a very large amount.
There was not a similar necessity for restricting the amount in
the ultimate portion of the tables because of the larger volume
of exposures during that period. It will also be noticed that
the limit of $100,000 might be exceeded even during the select
period through the issuance of policies for less than $50,000 at
the same age, but this was not a serious practical objection to
the method followed.

In this investigation it was decided to terminate the exposure
if the amount insured under a policy was increased or decreased.
This eliminated much labor and avoided complications regarding
the select period where such a change was made upon evidence
of insurability.

A life may pass out of observation by any one of four modes of
termination: (1) Existing; (2) Withdrawn; (3) Matured; (4)
Died. The exposure of a life terminates by "existing" when the
policy is in existence at the close of the period covered by the
experience. Lapses and surrenders are classified as withdrawn
All involuntary withdrawals such as expired term policies and
matured endowments are treated as matured. The withdrawn
and matured have often been grouped together, while on the
other hand additional classifications may be adopted depending
on whether the experience is to be used for additional information,
such as the rate of voluntary withdrawal.

In recording the exposures and deaths in a mortality experi-
ence three methods are available; the "Policy Year," "Calendar

14 CONSTRUCTION OF MORTALITY TABLES

Year," and "Life Year" methods. These terms apply to the
manner of analyzing the exposures and deaths.

A. Policy Year Method.

Under the policy year method the exposures are traced from
the beginning to the end of each policy year and each death is
allocated within the exact policy year in which it occurs. The
age at entry may be taken as the age nearest birthday, or it may
be taken as the mean age, found by subtracting the calendar
year of birth from the calendar year of issue. While in any
particular case the mean age may be nearly one year greater or
less than the correct age, these discrepancies may be considered
to balance. In the United States the nearest age method can
be followed by extracting the age directly from the policy records.
In Great Britian and Canada, however, it is customary to insure
at the age next birthday and either method would require a
calculation.

As a policy year investigation ordinarily begins and ends with
policy anniversaries in specified calendar years the existing will
usually pass out of observation at an integral age determined by
adding to the age at issue the duration found by subtracting the
year of issue from the year with which the experience closes.

In treating the withdrawn fractional durations are involved.
These may be treated as the "exact," "nearest," or "mean"
duration. The exact method is followed by tabulating the
precise fractions of a year of exposure over integral years (in
practice usually to the nearest month). The nearest duration
is found by takiag the nearest integral number of years, with
proper adjustment so that in cases where the fraction is | the
number of cases counted as the next higher integral year will
balance those counted as the next lower. The mean duration is
found by subtracting the year of entry from the year of exit on
the supposition that the overstated durations will approximately
balance those understated. The exact method is usually
laborious and one of the other methods is therefore to be pre-
ferred if it can be found to give approximately correct results.
The nearest duration method may in some instances materi-
ally understate the exposures; for example, in the first
policy year where the lapses at the end of three months may over-
balance the lapses at the end of nine months and thus affect the
exposures for the first policy year. The mean duration method is

FROM THE RECORDS OF INSURED LIVES. 16

based on the assumption of an even distribution of business
throughout the calendar year, which may or may not be suffi-
ciently in accordance with the facts in any particular investiga-
tion. In general it may be remarked that neither the nearest
nor the mean duration method should be adopted without taking
into consideration all the peculiarities of the fundamental data.

Those cases which pass out of observation through maturity
seldom introduce fractional durations, but if they arise fractions
may be treated in the same way as for the withdrawn.

Deaths must always be treated with accuracy as any error
would be of vast importance compared with a similar error in
the exposures. As a life is treated as exposed to risk during the
whole year of death the duration would naturally be taken to
the policy anniversary following death. The curtate (next
lower integral) duration, however, may be recorded provided
that, in tabulating the results, the exposures are adjusted to
include the year of death.

As the policy year method tabulates the experience in the form
which is desired for the construction of select tables it is the
method which would naturally be employed in important in-
vestigations of that character. It was used as the basis of the
British Offices' Experience (1863-1893), the Medico- Actuarial
Investigation (1885-1909), and the American-Canadian Mor-
tality Investigation (1900-1915). A full description of these
applications of the method may be found in the published
volumes of those experiences.

B. Calendar Year Method.

The calendar year method is an important historical method,
used first by Mr. Woolhouse in the 17 Offices' Experience, and
used later in the Institute (20 Offices') Experience and the Thirty
American Offices' Experience. It should be noted that this
method can be used where the policy details are not sufficiently
complete for a policy year method. In connection with its use
in the Institute Experience it was stated that much valuable
data could not have been included in the experience had precise
dates of birth, entry and exit been asked for. All that was
available in many cases was the office entry age next birthday
and the calendar years of entry and exit. Policy year tabulation
was therefore impossible.

16 CONSTRUCTION OF MORTALITY TABLES

The method has however lost favor as the basis of important
mortality tables constructed from life insurance experience be-
cause it does not produce data in convenient form for select
mortality tables. The material may be separated by calendar
years after entry and select tables compiled by an approximate
method as hereafter described; but year 0 — the calendar year
of entry — represents only approximately the first six months
experience and year 1 extends approximately from duration ^
to 1|, etc. Therefore, if reliable select tables are desired the
policy year method should be used. It should also be noted
that, under the assumption that those entering and withdrawing
do so in the middle of the year, persons withdrawing in the
calendar year of entry are excluded from the experience, thus
tending to understate the exposures.

Under the calendar year method of analyzing exposures and
deaths the calculations in connection with the exposures are
usually based upon the assumption of an even distribution of
issues throughout the year. Thus new entrants may be assumed
to be exposed on the average for six months in the year in which
the policies are issued and similarly withdrawals can be con-
sidered as taking place in the middle of the year of exit.

The new entrant is assumed to enter in the middle of the
calendar year, which is the period used as the basis for grouping
the exposures. In this first calendar year therefore he will be
treated as exposed to risk for one half a year at his age at the
beginning of such calendar year. He will be counted as exposed
for the next calendar year at his age at the beginning of that
year. Therefore, if the age at date of entry were calculated as
the age nearest birthday or mean age at that date, he would
enter as a fractional exposure for that year at an age 5 year less
and would be exposed throughout the next year at an age | year
more. So if the age at date of entry were calculated as an in-
teger, X, the calendar year exposures would be at the fractional
ages X — ^, X -\- i, X -{- 3/2, etc. The deaths would also have
to be tabulated by half ages. The completed age at death would
be the half age on the January 1st preceding death. We may
find the completed age by adding the proper duration to the age
at entry. If death occurs in the first calendar year following the
year of entry we should add ^ to that age, if death occurs during
the second year after the year of entry we should add I-2-, etc.
Thus the general rule for determining the completed age would

FROM THE RECORDS OF INSURED LIVES. 17

be to add to the age at entry (nearest birthday or mean age)
the mean duration minus one-half. For example, if an insured
were 35 at the date of issue in 1890 and died in 1912, his age on
January 1, 1912 would be 35+ (1912 - 1890) - h, or 56^ In
this way the exposures and deaths would be grouped by half
ages and from the values of qx+i so obtained the values of q^
might be interpolated.

It would seem preferable in many cases to derive the rates of
mortality directly for integral ages, but in order to accomplish
this we must follow a method which will assume that integral
ages among the exposed to risk coincide with January 1st.
Where the practice of insuring lives at the age next birthday
exists, it is convenient to consider that such ages will be attained
on the January 1st following entry. This is equivalent to the
assumption that the age next birthday is on the average | year
greater than the exact age at date of entry and that on the
average six months elapses between the date of entry and the be-
ginning of the next calendar year. Each life will then be exposed
to risk for ^ year in the calendar year of entry at an integral
age one year less than the age next birthday at issue, and the
succeeding years of exposure will obviously be at integral ages
also. Withdrawals would be exposed for | year at the integral
age corresponding to January 1st of the calendar year of with-
drawal.*

Another method that will group the data by integral ages
consists of using as the age on January 1st following issue the
age nearest birthday on January 1st. If the age so calculated
is X, the life would be exposed for half a year at age x — 1 in the
calendar year of entry.

If the exposures were to be tabulated for integral ages by
calculating the age nearest birthday on January 1st following
the date of entry or by using as the age on such January 1st the
age next birthday at entry, then the age on January 1st of the
year of death may be taken by adding to such an age the mean
duration minus one. This gives precisely the same result in
the former case as would be obtained by calculating the age

* See Mr. Geo. King's remarks, J. I. A., Vol. XXVII (pg. 218) for alternate
method of treating new entrants and withdrawals. It should be remembered
that the fractional exposures in the years of entry and withdrawal might be
calculated accxirately but such refinement would destroy the convenience of
the method.

18 CONSTRUCTION OF MORTALITY TABLES

nearest birthday at the beginning of the year of death. In
the latter case the deaths would ordinarily be given somewhat
more accurately by calculating the age nearest birthday on the
January 1st preceding death instead of employing the rule
just given, which uses the age next birthday at issue. It must
be remembered that whatever method of determining the age
is used, each death must be included as an exposure for the
entire calendar year of death.

For the withdrawn, the age at the beginning of the year of
withdrawal may be determined in accordance with the con-
siderations just mentioned. If the age next birthday at issue
was used, however, it would be satisfactory to add the mean
duration minus one rather than to calculate the age nearest
birthday on January 1st of the year of withdrawal as is some-
times done for deaths.

C. Life Year Method.

The third way of analyzing the exposures and deaths is the
life year method. This might be called the exact method as it
traces the data from birthday to birthday. The deaths are
grouped according to the exact age last birthday at the date of
death. The exact ages (to nearest month, or to one decimal
place) at entry and exit of all lives entering into the experience
may be calculated, each life being ordinarily exposed for a frac-
tion of a year of age at the beginning and end of its exposure.
Instead, however, of calculating each of these fractions with
exactness an average fraction may be assumed, after a sufficient
test, to apply to all cases except deaths, which are treated as
exposed to risk for the whole year of death.

In experiences where it is justified a rougher approximation
may be adopted for computing the exposed. This consists in
taking the ages at entry and withdrawal as the mean ages found
by subtracting the calendar year of birth from the calendar
year of entry and exit respectively. This avoids fractional
durations and is based on the assumption that the resulting
over-exposures and under-exposures will balance. There is a
difficulty presented if the experience starts at a certain date
with a number of lives then under observation or closes with a
number existing. For such cases the ages at entrance and exit
may be taken as the age nearest birthday or determined by some
other suitable method.

FROM THE RECORDS OF INSURED LIVES. 19

Another method of estimating the exposures is available by-
using the ages nearest birthday at entry and exit for the entire
data except deaths. Under this method the error involved in
the age at entry or exit for any one life cannot exceed six
months, while the mean age may be incorrect by almost a year;
but under both methods the largest possible error in duration is
approximately one year.

The life year method, while making possible the strictest
accuracy if the ages are exactly calculated, has a very narrow
scope of usefulness in life insurance experience, because it does
not produce the results in the form of select tables and, if applied
with exactness, is laborious. It has been employed in tabulating
mortality among annuitants.*

A general warning should be given regarding the use of ap-
proximations which have been frequently suggested in connection
with all these methods. In any important investigation the
effects of such approximations should be tested before they are
adopted. For example, where an office accepts applicants at
rates for the nearest birthday there will be a tendency for persons
to insure just before the nearest birthday changes. In such
case the nearest birthday at entry would on the average some-
what understate the true age.f If lives are accepted at rates
for the age next birthday the tendency will be to insure more
often just before than just after a birthday, and the result of
using the nearest age may be to overstate the age slightly.
Again a company's business may not be evenly distributed over
the year, because of pressure at the end of its financial year or
toward the culmination of agency contests, and methods de-
pending upon such even distribution may produce too great an
error. Considerations such as these should be taken into account
in determining the methods to use.

Mortality investigations frequently cover the experience under
all policies in force within a specified period. It then happens
that many lives will enter the experience at the beginning of
the period after having been insured for some time.

* For applications of the method see J. I. A., Vol. XXXI, pages 171-5
and 179-181.

t If a mortality experience is to be used as a basis for premium rates charged
at nearest ages there is no objection to such understatement of ages in the
data with resulting slight increase in mortality rates; for the companies will
thus have provision in the mortaUty table for the effect of charging premiums
at nearest age.

20 CONSTRUCTION OF MORTALITY TABLES

If the policy year method is to be used the experience is gen-
erally taken between the policy anniversaries in the first and
last years of the period. By this means fractional exposures are
avoided at the beginning in the case of lives brought forward.
Care must be taken however to see that the policy years elapsed
since original entry are properly recorded for use in tabulating
select tables. If other methods are used the manner of handling
the lives that enter the experience in this way must be consistent
with that employed for lives first insured during the period of
investigation.

It is customary to allow policyholders a grace of thirty days or
one month for the payment of renewal premiums. If death
occurs during the grace period while the premium is unpaid the
claim is allowed, the overdue premium being deducted from the
face of the policy.. As such deaths will be recorded in a mortality
experience theoretical accuracy might seem to require treating
every lapsed policy as exposed to risk for one month after the
due date of the premium, in order that the corresponding ex-
posures may be recorded; or, if these fractional exposures are
not taken into account, then it might appear that no account
should be taken of the corresponding deaths.

It is impossible, however, to ascertain what deaths should
properly be eliminated under the latter alternative because we
ought to exclude only those cases where the premiums would
not have been paid had the person insured survived the grace
period. Furthermore there are practical objections to treating
the grace period on lapsed policies as exposures, because com-
panies will continue to grant a grace privilege and their premiums
must be suflficient to provide that benefit. By including the
deaths and excluding these fractional exposures the mortality
table will give premiums which contain a provision for this.
It is therefore the usual practice to include the deaths and ex-
clude the exposures.*

When a policy lapses after having been carried a short time,
it is customary to grant an automatic nonforfeiture feature.
If automatic extended insurance is given then it must be de-
cided whether the exposure is to cover the period of risk under
the original form of policy only, or whether the risk under the

* The British Offices' Experience is an important exception to this rule.
The nearest duration method was modified to take account of the grace period.
See Appendix M, volume entitled "Account of Principles and Methods."

FROM THE RECORDS OF INSURED LIVES. 21

term extension is to be included. When the decision is made the
deaths are taken to correspond. If the risk under the extended
term insurance is to be included the fractional durations at the
expiry of the term insurance must be treated in the same way
as the fractional exposures of the withdrawals.

Many policies are changed at lapse to reduced paid-up in-
surance either automatically or upon request. If the experience
is based upon amounts insured, such policies, if included in the
experience, must be treated as withdrawals for the amount of
the reduction and the balance continued. This is awkward and
it may be thought best to exclude all data from the date of lapse.

It must also be kept in mind in treating paid-up and extended
insurance that the character of the mortality under these options
may vary considerably from that under policies in full force.
It may therefore be best in some circumstances to make separate
investigations for such lapsed policies to determine the company's
attitude toward the non-forfeiture features and to exclude the
data from the company's general experience.

The question also arises whether to include policies after the
amount of insurance has been changed. Except where the
investigation is by amounts iasured a reduction in amount
presents no diiO&culties, but as an increased insurance is usually
granted only upon medical examination such lives would be
select from the date of change. If an aggregate table is to be
constructed this might be disregarded but for a select table
recognition must be taken of this fact. Because of such com-
plications it will sometimes be best to exclude one or both of these
classes of changed policies.

The same principles will control the decision as to the inclusion
of policies changed in form, for changes to lower premium forms
are made subject to examination.

If a policy lapses and is afterwards reinstated upon evidence of
good health it must be decided whether the exposure should
terminate at the date of lapse and start again at the date of
reinstatement, or whether the temporary lapse should be dis-
regarded. Where the period between the date of lapse and date
of reinstatement is short and only a personal certificate of health
has been furnished, no appreciable error will be introduced by
ignoring the lapse and reinstatement. Where considerable time
elapses over which the policy is not exposed to risk and a new
medical examination is made, it may be thought better to take

22 CONSTRUCTION OF MORTALITY TABLES.

account of the lapse and terminate the exposure at that point.
The subsequent reinstatement may then be treated as a new
policy; but, bearing in mind that a company is not always so
particular in reinstating a lapsed policy as in issuing a new one,
and often reinstates an impaired life covered by extended in-
surance because the mortality loss will be less under the premium-
paying policy, it may be thought better to eliminate entirely
from the investigation the exposures under such reinstated
policies. It is not likely that any material effect upon the results
of the investigation would be caused either by including or ex-
cluding these exposures.

In the construction of mortality tables many points will arise
which call for the independent judgment of the compiler, and
when the tables are published it is usual to state how these
different points have been handled.

When constructing such tables it is sometimes desired to
investigate collateral questions such as the relative mortality
by plan of insurance, amount of insurance, residence, occupation
or other features of the risk. It is therefore necessary to plan
the card on which the policy details are to be recorded so that
provision will be made for the information necessary for such
investigations.

Chapter IV. Handling the Data.

The compilation of a mortality experience is usually carried
out, as has been mentioned, by recording the data on cards
because of the ease of sorting the material by age at issue, dura-
tion, mode of exit or any other feature. It is necessary to
exercise care to see that the sources from which these details
are drawn are reliable, particularly in regard to the mode of
exit; for it is essential that the deaths be recorded correctly.
It may be worth while to check all cases recorded as dead with
the claim records, if they are available. If the cause of death
is to be used for a subsequent investigation of the relative
frequency of the different causes, such a check may be obtained
with little or no extra labor.

If the table to be constructed is an aggregate table or simply
excludes certain early years of duration, as in the case of a
truncated table, it is necessary to record the age at entrance
into the experience and at exit. Equivalent facts would be
obtained by recording the age at entry and duration, from which
the age at exit can be obtained by addition. When select tables
are desired it is necessary to show the duration instead of the
age at exit.

In writing the cards the policy records must be examined care-
fully to exclude the data which is not desired, as for example,
under-average lives, female lives, policies continued under non-
forfeiture provisions, residents outside the temperate zone, etc.

If the mortality table is to be based on lives insured, the date of
birth and name, or all initials, should be entered so that dupli-
cates may be eliminated as described in Chapter III. For a
table based on amounts insured care must be taken to record
properly cases in which a change in the amount of insurance has
occurred. If such cases are to be included in the experience
after the date of change it will probably be best to write two
cards, one for the lesser amount and one for the balance, with
proper dates of entrance and withdrawal to make them equivalent
to the case as a whole. When an experience both by policies
and by amounts is desired, the lesser amount may be counted as
one policy while the balance may be indicated as "no policy."

24 CONSTRUCTION OF MORTALITY TABLES

Then in combination they will count as one policy for the full
amount.

For select tables cases re-examined medically, such as in-
creased amounts, changes to lower premium forms, and possibly
reinstatements, if included in the investigation, would usually
be recorded as withdrawn at the time of change or lapse, and
additional cards would be written to tabulate them as new
entrants after such dates.

Policies which enter the experience at the beginning of the
period covered by the investigation, having been previously
issued, must be specially treated. If only an aggregate table is
desired the age at entry may be recorded as the age at date of
entering the period. If, however, select tables are to be con-
structed the card must show the age at issue, duration at entry
into the experience, and duration at exit.

Those policies which are still in effect at the close of the period
of investigation will have the duration at that time recorded.

The form of card to be used should be selected with care to
include all information necessary to permit of the easiest and
most accurate recording, and to make the elimination of dupli-
cates, sorting and tabulating as rapid as possible. The volume
entitled "Account of Principles and Methods" of the British
OflSces' Experience (1863-1893) gives an excellent example of
the points which must be considered. The card shown on the
following page is one of several forms used in that experience.

The letters S.A. indicate that this card was to be included in
the data both for the select and also for the aggregate tables.
The age at entry was taken as the age nearest birthday. There-
fore + was marked after the year of birth in this individual case
to indicate that the age at entry was the difference between the
year of entry and one more than the year of birth. The data
were analyzed by policy years, and the duration recorded on the
cards for deaths was the curtate duration, or the duration at
the commencement of the policy year of death. Cards of
different color were used for male and female lives.

It is interesting to note that a preliminary test showed that a
general alphabetical arrangement of the cards would not satis-
factorily eliminate duplicates. The cards were therefore sorted
chronologically by date of birth. After noting the duplicates
detected in this way, the cards in each year of birth were sorted
alphabetically and again examined for duplicates.

PROM THE RECORDS OF INSURED LIVES.

S.A.
NO 2356

Old Policies. N^^
£....100.

CLASS....0

PROFIT OR NOT....P

(Latimer ...

LIFE \

\ Darsie

DATE—

OF BIRTH

M. Year.
8. 1826 +

OF ENTRY...

1 1850

In 1863

1863.

OF EXIT

2 1888

Duration before 1863

13.

Duration of Policy.

38......

Age at Entry

Age in 1863

Age at Exit

MODE OF EXIT.

(D)

REMARKS.

J. ^^

(Select and Aggregate Tables.)

In compiling the census returns of 1890, the United States
Census Office made the first use of the system of punched cards,
which was devised by Dr. Hollerith and permits of mechanical
sorting and tabulating. This system makes use of a card
printed with a series of columns running from 0 to 9. If the
card is to represent a policy issued at age 35, the "3" in the
tens column and the "5" in the units column representing the
age will be punched out. By means of code numbers the occupa-

26 CONSTRUCTION OF MORTALITY TABLES

tion or other information may be entered upon the Hollerith
cards; for example, two columns will cover 100 and three columns
1,000 different occupations. By an ingenious mechanical device
the capacity of the cards may be somewhat increased beyond the
100 or the 1,000 without increasing the number of the columns
beyond the two or three. After the cards are punched they
are sorted mechanically at a high rate of speed, it being claimed
that 15,000 cards may be sorted in one hour. Another machine
specially designed will add four items on the cards at a speed
slightly lower than that of the sorting machine. Machines may
be built on the same lines to add different numbers of items.
The student is referred to two articles (T. A. S. A.,X1, 252, 276)
for further information concerning the Hollerith system.

Another system which employs similar punched cards is that
put out by the Powers Accounting Machine Company. These
machines were used in carrying out the American-Canadian
Mortality Investigation. A copy of the card used in that ex-
perience is given on page 27. (See Page 11, A. C. M. I.)

Volume 1 of the investigation states "The machines used for
perforating the cards were so constructed that the cards auto-
matically came into place for perforation. The hole to be
punched in each column of the cards is selected by a sliding
scale or bar and the perioration is done by electricity. Two
cards may be prepared at one time, and accordingly, cards of
two colors are inserted alternately in the package of blank cards
and placed in the machine. The verification of the accuracy
in perforating the cards may be done in two ways, either by
comparing the perforated card with the original record, or by
using a verifying machine so constructed that when the operator
strikes an incorrect key or when the key struck does not agree
with the hole already in the card the machine automatically
stops and thus attention is called to an error on the part of the
original perforating clerk or of the operator of the verifying
machine.

"A great advantage of this perforating machine is that in certain types of
perforation the keys need not be changed if the same figures appear in a
number of cards. If, for example, the cards were all issues of the year 1914
and were issued in the State of New York on the twenty year endowment
insurance plan, then in the proper fields the figiires 14, 3 and 38 would remain
automatically in place until all the cards coming within that group had been
perforated. Another advantage is that the complete setting of the machine
for perforation can be read along the edge of the machine before touching the
lever which perforates the card."

FROM THE RECORDS OF INSURED LIVES.

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28 CONSTRUCTION OF MORTALITY TABLES

The following statements regarding the sorting and tabulating
machines will give a good idea of the remaining operations of
this system.

"The sorting machine if kept suppUed with cards, will classify 12,000 per
hour. This, of course, refers to sorting any one column. If the cards are
to be sorted by age, two sortings are necessary, — first, into groups of decennial
ages, i.e., 20-29, 30-39, etc.; and then into separate age groups. Advantages
of this type of machine are that the cards are placed by machine in compart-
ments which are easy to empty, as they do not involve stooping, and that
all cards of a specified number can be segregated, leaving the cards with other
nimabers in the same numerical or other order. Several machines used by
the bureau had counters, technically known as coimting sorters, attached to
the sorting machines. These machines count the number of cards going into
each compartment or pocket and give also the total number of cards in all
the pockets.

"This machine (the tabulator) prints on strips of paper such fields in the
cards as are desired. This is known as selecting mechanically the desired
coliunns of the cards to be recorded on the list. It can be so arranged as to
record letters instead of the figures appearing on the card. For example, the
code number for the termination of the policy by death is 9, but it appears on
the list prepared by the machine as ' D.' From one to seventeen columns may
be added. The work of this machine can best be imderstood by means of
the following example:

Example of Work bt Tabttlator-Printer.

Explanation of Columns of Figures or Symbols.
(o) Year of issue; i.e., 2 stands for 1902, 12 for 1912 and 8 for 1908.
(6) Age at entry.

(d) Sex (M = Men).

(e) Mode of termination (L = Lapsed).
(/) Code number for plan of insurance,

(<g) Code number for habitat at date of application for insurance.

(h) Nmnber of policies.

0) Amoimt of insurance to nearest $100; e.g., 10 = $1,000, and 25 = $2,500

(a) (6)

(c)

(d)

(e)

(/)

(9)

(h)

2 20

12 20

150

13 20

11 20

100

8 20

10 20

1 20

6 20

11 20

300

(Total niunber of policies) 12 (for ins. of) 730

[$73,000]

FROM THE RECORDS OF INSURED LIVES. 29

"The parts in brackets are not printed by the machine but are added for
the sake of clearness.

"After the cards have been sorted into the desired groups, special cards
known as 'total' cards were placed between these groups. The machine con-
tinues to print as shown above until it comes to a 'total' card when the desired
summations are made; the machine then commences to tabulate the ensuing
group. After all the cards in the machine have been hsted and added, the
machine automatically stops. An operator can handle two or more machines,
as it is necessary merely to keep them supplied with cards. Cards pass
through the machines at the rate of 2,500 per hour when there are large groups
and about 2,000 per hour when there are small groups necessitating frequent
summation.

" A study of the foregoing example of listing will indicate that the Tabulator-
Printer is of great value in detecting errors. In the example given, the insured
were all men of twenty years of age who had lapsed their poUcies with a diu-a-
tion of one year and were residing in State No. 16 (Illinois). Any errors re-
sulting from putting cards wrongly in any of these categories would be easily
detected. The same would hold true in other sortings, such as those among
women who had died in the third insurance year. In that case column (c)
should be ' 3 ' ; column (d) should be ' F ' ; and column (e) shovild be ' D.' Again,
if the group consisted of men aged 25 whose poUcies were existing at the close
of the investigation and who had taken their poUcies in 1905 on the twenty
payment life plan when they were resident in Massachusetts, column (a)
should be '5'; column (6) should consist only of '25'; column (d) of 'M';
column (e) of 'E'; column (/) of '2'; column (g) of '25'; column (c) should
be blank."

From the above descriptions of the mechanical systems in
use it will be seen that their invention has made it possible to
carry out expeditiously large mortality investigations which
would involve extraordinary effort if carried out by hand. The
successful operation of these systems, however, requires careful
planning of every detail that is to be recorded.

In connection with these mechanical systems it should be
kept in mind that in most cases it would be impracticable to
punch the cards directly from the company's records. It is
customary, therefore, to prepare preliminary sheets on which
can be written the data necessary for the cards. The duration
or age at exit may be calculated from the original records by
the person making out the working sheet, thus reducing the
punching of the cards to a simple and rapid operation. The
form of sheet used in the American-Canadian Mortality In-
vestigation is shown on the following page.

This form should be compared with the specimen card which
is shown on page 27 and which was punched from it.

CONSTRUCTION OF MORTALITY TABLES

«> a >•

a a

O g -tJ

a «S5»

«^ o) «;
® c3 g

3
03

o§

•aopnuTOuax

JO 8poj\[

t'"

o S

t— (

•nOl^BJUQ

«»J

"J a

Pi]

:= o

FROM THE RECORDS OF INSURED LIVES. 31

In investigations carried out by means of written cards, no
preliminary working sheets would ordinarily be necessary, as
the cards could be written directly from the original records.
It should be noted how the card on page 25, used in the British
Offices' Experience, is made up conveniently for copying from
the original records and then for computing the durations and
ages.

Before tabulating the cards, whether they be on the written or
punched form, it is desirable to have some rough check on their
accuracy even though they have already been checked individu-
ally with the source from which the data were drawn. This is
particularly advisable if the records of several companies are
being combined. In the American-Canadian Mortality In-
vestigation it is stated :

"It seemed essential to have some check on the data so as to determine
whether a section of the experience relating to existing, lapsed or deaths had
been omitted. So far as concerns the principal companies which furnished
their data on perforated cards, very careful checks were made by these com-
panies, including calculation of expected deaths by a standard table and
comparison with the actual death losses. For the other companies the bureau
tabulated the existing, discontinued and dead, according to age, and ascer-
tained if they were reasonable. The death losses were also subdivided by
insurance years and comparison made between the groups by ages and in-
surance years. This would probably have brought out any material error.

"All cases were investigated which showed that the age of entry was over
70 as the niunber of such might be few and therefore errors in perforating
might affect the results. Where the age at death was over 100 the company
reporting the case was asked to verify the original record."

Chapter V. Tabulating the Data.

Before sorting the cards on which the details have been re-
corded, careful plans should be made for obtaining the exposed
to risk in the easiest manner possible. A continuous process is
used by which Ex+i, the exposed to risk at age x -\- 1, is derived
from Ex. The formula to be employed will depend upon the
method of analyzing the exposures and deaths — policy, calendar,
or life year method — and upon the character of the mortality
table to be constructed.*

A. Policy Year Method.

Let us first consider the construction of an aggregate table by
the policy year method.

Let rix = New entrants at age x.

ex = Existing at age x at close of observations.

ffx = Those under observation at age x when the observa-
tions began.

Wx = Withdrawals at completed age x.

Qx = Aggregate of fractional durations beyond com-
pleted age X arising from the Wx withdrawals.

dx = Deaths occurring between ages x and x + I.

Let it be assumed that Wx includes both the withdrawals and
the matured. If it is desirable to have them separated for
other purposes, separate symbols may be assigned to each, but
the formulae will hold with the corresponding substitutions.
It will be noticed that the completed age at death is used, i.e.,
the age at issue plus the curtate duration. Therefore the dx
deaths must be treated as exposed to risk to age x + 1.

Then

XXX X X— 1

Ex = E^x + Z<^x - T.Wx + Qx - E«x - E^x
0 0 U U 0

and

Ex+i = Ex -\- rix+i + Cx+i — 'M'x+i + (^x+i — ^i) — ex+i — Ox. (1)

* The authors wish to acknowledge their indebtedness to Mr. Robert
Henderson for his concise explanation of formulae and methods of treating
ages and diu'ations contained in his monograph "Mortality Laws and Statis-
tics" published by John Wiley & Sons, Inc.

FROM THE RECORDS OF INSURED LIVES.

It will be seen that the use of gx assumes that the exact dura-
tion method has been used. If nearest or mean durations are
employed the withdrawals take place at integral ages and Qx
becomes zero for all values of x.

In order to carry out this formula, the information may be
recorded on one sheet similar to the following.

(1)
New
En-
trants.

(2)
Enter-
ing
Obser-
vation.

(3) =

(1)
+ (2).

(4)
With-
draw-
als.

(5,

Exist-
ing.

(6)
Dead.

(7) =
(4) -|-(5)
-F(6) Pre-
vious line.

(8)

Sx=(3)

-(7) +(8)

line.

(9)

The cards would be sorted by the mode of entrance corre-
sponding to columns (1) and (2), if they were not already so
separated because of their different years of issue. Each set
of cards would then be sorted by age at entrance into the ob-
servations and the totals for each age recorded. The next step
would be to sort them all in accordance with the mode of exit and
then by age at exit. In tabulating the number of withdrawals
at age x we shall have the sum of the fractional exposures, gx,
to tabulate also if the exact duration method was used. If,
however, we tabulate the expression {wx + gx-i — gx) in column
(4), the value of column (7) will be {wx + gx-\ — gx -{- ex + 6x-i),
which is the quantity that enters negatively in equation (1).

This method of tabulating might be described as assuming
that, of the Wx withdrawals at completed age x, we consider gx
as withdrawn at age x + 1 and Wx — gx at age x. A separate
sheet or column might be provided for recording the values of
gx and of gx-i — gx- It will be noticed that this latter value may
be negative, in which cases its absolute value is deducted from Wx.

Where select tables are to be constructed the exposed to risk
must be tabulated for each age at issue separately and the
formulae become

Eix]+t = nix] + T,<^[x]+t — Zw[x]+t + gix\-\-t

1 0

t t-\

1 u

CONSTRUCTION OF MORTALITY TABLES

and

Eix]+t+i = E[x]+t + o-[x]+t+i

— W[x]+t+i + (gix]+i+i — giz]+t) — e[x]+t+i — 6[x]+t (2)

where exact durations have been used.

If the nearest or mean durations are taken, gix]+t vanishes and
we have

E[x]+t+i = -£^[i]+t + <r[x]+t+i — (w[g]+t+i + €ix]+t+i + d[x]+t) (3)

In all of the foregoing equations it has been assumed that
observations began and ended on policy anniversaries in the
first and last calendar years of the investigation, so that there
are no fractional durations for a^ and e,; otherwise the fractional
exposures must be treated in a manner similar to those arising
from the withdrawals.

In order to produce the values of E[x]+t+i by equation (3) it
will be seen that the sorted cards may be tabulated on sheets
drawn up somewhat as follows, using a separate sheet for each
age at issue.

Age at Issue

Number of Kntrant«

Dura-
tion.

(1)
Entering
Observa-
tions.

(2)

With-
drawals.

(3)
Existing.

(4)
Dead.

(5)

Decrement

(2)-H(3)-(l)

-h Previous

Line (4).

(6)

Exposed to Risk

(6) Previous

Line —(5).

The exposed to risk at duration 0 would be found by sub-
tracting the withdrawals at that duration from the number of
entrants shown at the top of the sheet. There would naturally
be none entering the observations or existing at duration 0.

To have the cards in the proper order, they must be sorted
by age at issue, then by mode of exit and finally by duration at
exit. The cards for <Tix]+t must also be sorted by duration at
entrance into the experience.

FROM THE RECORDS OF INSURED LIVES. 35

If the exact duration method is used for the withdrawals the
cards should show and be sorted by the curtate duration. For
each duration there must then be calculated the value of g[x]+t-
This number may be tabulated as withdrawing at duration t -f 1
and the balance of the withdrawals, (w[x]+t — g[x]+t), may be
placed opposite the curtate duration, t. This will permit
Eix]+t to be computed in the same way as where mean or nearest
durations were employed. In writing down the withdrawals
on the sheet two entries will be made for each duration, except
duration 0, because of the withdrawals thrown forward and those
thrown back. It would therefore be well to have the column
for withdrawals subdivided for each duration by a light horizontal
line.

Having thus obtained the exposed to risk the select mortality
rates may be computed, for

Ql']+t

w+<

The period of selection may also be determined by combining the
data on" the sheets for the several ages at entry as described in
Chapter I. Where an aggregate table is wanted in addition to
select tables, the deaths and exposed to risk for each attained
age may be footed from the sheets and the ratios of these totals
calculated.

It may be desired to obtain an ultimate or a truncated table
only. If we let t be the maximum duration excluded from such
an experience then every policy in force not more than t years
would be eliminated and every other policy would be recorded
as entering at age x -\- t. In other respects the same method
could be followed and the same formulae used as for an aggregate
table. To construct a true ultimate table, however, it would
not be possible to determine t beforehand and it would be neces-
sary to list the select data for that purpose. The method just
mentioned is therefore practicable only for a truncated table
where the experience to be excluded is fixed before the investiga-
tion is tabulated.

In the British Offices' Assurance Experience the withdrawals,
that is lapses and surrenders, were designated by iy[x]+«, while
the matured, called "Terminations," were treated separately
under the symbol 7'[i]+«. For both classes the nearest duration
method was used, properly modified to include the days of grace

36 CONSTRUCTION OF MORTALITY TABLES

in the exposures. The value of W[x]+t was the sum of W^''\x]+t-i
and W'-''\x]+t, namely the withdrawals in the tth and (t + l)th
policy years that were taken as withdrawing at integral duration
t. Similarly Tix]+t represents the sum of ^(^[xi+t-i and T^''\x]+f
The symbol Gix]+t was used to represent the net movement,
i.e., those entering the exposures at duration t less those emerging.
Therefore Gix]+t = cui+t — {d[xu-t-i + wm+i + T[x]+i + eix]+t)
and, as a[x] was used to represent n[x] so that G^x] = o-[»i — it>[ij,

Eix]+t = E[x]+t-i + Gix]+t or Eix]+t = IC^w+t-

A specimen working sheet is shown on page 37.

This table contains the data only for "New" assurances,
namely, issues from 1863 on, and therefore (rix]+t has a value
only at duration 0, representing new issues.

In combining the select data for the construction of an aggre-
gate table it was necessary to sum the several values for the same
attained age. Thus Wx = W[o]+x + if [u+x-i + W(2i+x-2 + • • •
+ W[x-i]+i + W[x]+o and similarly for 6x, <Tx, Tx and e^. Then
Gx = <Tx- (dx-i + Wx-}- Tx-h ex).

The corresponding working sheet is given on page 38.

For the construction of truncated aggregate tables the follow-
ing equation was used,

^^(«) = Ex — iE[x] + E[x-i]+i + • • • + Eix-t+i]+t-i),

where Ex^^^ indicates the exposures at attained age x excluding
the first t years of duration.

The American-Canadian Mortality Investigation furnishes an
interesting example of a working sheet combining policies
issued both before and during the period of investigation. The
experience was based on amounts insured to the nearest hundred
dollars. Mean durations were used (see page 39).

The years of issue shown in column (2) apply to the Existing
in column (3). Similarly column (7) refers to the Entered in
column (8) These years were inserted to make the tabulation
as simple as possible. Column (10) was obtained by summing
column (9) from the bottom up instead of working from the top
down in the usual manner by subtraction. The text of this
investigation gives the formula

^[;r] + n = E[x]+n-l — (e + W + d — s)[x] + n,

FROM THE RECORDS OF INSURED LIVES.

•-J

•f3

m
in

8
o

-a

fa
o

S o «

O P QD

1-1

l-H

1-1
o

i-icOt^t^OMOCOlMM
«D «3 «^ (N t<^ CO O <C ■*_ (N

~Wo

t»

«OT)<coeoNC<Mi-trt»-*

OlOQOOi-iWCOWTtiOS

■"fOJt-oiOTjieowiNiH

1-1

►2 a

1-1

s
1^

1-1

•

^(O oo

3fl

+ J

iMOiOQ.-iiO(N(NMiN
0"5000<-iMCOeOTj(05

o S

2 4-Oa

1-1

Tt<OJt>.coio-^«eo(Ni-i

/-N»0 t»

S+ +

d w

00Ob»05O«0(Nb.OO

OiN00cO(N>Offlt^'-iN.

(^.9

SJ"

00<O>OTj<Tj<e0(N(N<Ni-t

-a

JJ,

'-"

-a

•H

d
'■5

d
o

■§

•

»o

r-(O-*Q0C0-<J(.-((Nt^O

M050<ooe<5eo<Ni-4i-i

-3

1
5

1-1

Tt< rH ,-(

d
o

1
+

t^M'*Q0(NO(3505i-ii0

i
1

2i-

■>!j<(Nt-M<NCOr-Hi-ii-i

iH i-(

Ot*O>0C0i0O-<i<i0t^

i-iiH(N(Ni-lT-tcqrH

■*»

<«>

^•1

1-1

■^
M

S-

rH(N«i}H>OO^iOOOO

CONSTRUCTION OF MORTALITY TABLES

T-t

• fe

-^-s

»5

-o

a
o

g-^«

«t5

«>43

1'^

^fS

•^ <

PL,

Exposed

Risk.

t^-*^j^wooow»a5!

v_^

iO«OC0«i2MNM»HC>d»

-Wo

>-l

C/J

oot^ooeooooocooo-»»«t>-

•HOOiOMCDTjt^Csr^tO

s
1

o
+

CO<N(NNNMN»-<^'-'

00
00

•»>

>-*N y-A

— -o 00

o »

^ i
p

TJH v.^^^

+ .

+ + +

^ w

0»0'«}<060S'«'05»01^

»H

N00t0e«5OTl<'-<05t»O

.— s«0 t^

roNe<ieMeM(NN'-i'-t<-t

CO •-'N^

""++

1§

t-«05(NMO(3>'Hl^«0

W 2-

»-i^OOSlN^t^l>Tt<fO

^.s

(NNNrHNfN'-l'-l'-l'-l

.... C4 ... ^ C4

r— 1

^.^

•

■a

•

'it

-a o

§g

1>H

o
*43

■<

|z]

t*QOOSeO<COC<J(Ni-«^

•§

«o

t-<

t»CO(N(NMe><— ii-i^^

f-<

^■^^

Nt»Tj<^ooasiot*os

"S p.

C<l N 1-H 1-1 1-1 .H

t-l
1

Tj<.-<ooooosa>'^rtOJO>

■*•

«>

l§

e<i(N(N-H^,.i<,-i,-(^ .

■^5

»-i

»-|<N^^■^lO;o^.«o>o

FROM THE RECORDS OF INSURED LIVES.

« 2

s ;^

0 o §.2

go a«

m OH

^ OS'S

00 ^-'

« g

I O <— ' t* 05 »0 05 CD '

OOOOOpO-i
OOOOOOOOOO

QO 00 <N t'
00 ^ 00 IN
t^ 00 rH t>.

O rH 00 (N

rjt CO 1-1 (N

OS_ rH N W lO 00_ r)H_ N lO O

CO OS* "f lO •* O" lO >o oT oo"
•>*(M005.-cOiOOOt^CO
FHT)<_iO(NCOiOCOt^T)<_CO

t>r (N oT tCio CO i-T oT oT Qo"
<o«OTjtTi<Ti(-»}<Tj<cococo

^COOOOiOtOO

000St^Ot>'^(Nt>^05_Q0

•^" CO o i-T t>r ^■' o ic o" 05

^iMOOOOiOt-Oi-HO

t^05e<o>oo»-j^»oeo>-'«D

CO rH

lO CO t> rH U3

Ti(i-ir»co-<i<^i>i>t^(Noo

N Tl^" O" '* -** CO |C CO (N 5D

Tj4-.i<,-i»oooeoooo>M*to
t>«OTtrcocoe<fi-re<fe4"i-J"

« o o

CO oiu3

OSOOt»<©iOTt<COiM^O

00000000000000000000

ooot»«oiCTj<eo(Mr-(o

t^ratfoioNodb^oTeoto

lOcDINCOOMiOOiOCO

1-H 1-H i-H b- 0_ t>^ lO lO rH (©

(N O) i> ui" lO -"jJ" CO* (N CO TjT

oeoooujoioo
lOoTioioeot^i-rio

eO_ <N "5 O (N l> IN O CO (N

odoco-^co'cTr-TiCodiC

(^(NINiNlNCOOOCOCOTj*

«DrtcO0U5-<}<«D(N«DlN00
«D 00 oT ^* O" cJ 00 IN o" O

OOr-^C0Oi-H-<}<Ot^O

0)OiNO«Dt)<^(NOO-«}<
lO" CO (N C<f <-H ^ rH <N

CO T)J^ tH 00 iC 00 »0 <N O O

<NO)t^o)o"ioooo)odoo

O'^TtiiNOM'COt^iOtO

o__ o !0 Tj^^ i-H o o CO 00 t^

»5 "5 •<* CO Co" CO rt" l-T rH r-T

TjHCOlNrHOOOOt^OlO

OOOOOOOOOO

T-i(NCOTj*iO!Dt>0000

U3 . IO_ 0_

eo ■ i-Tio

o_ o_ o_

i»coi-r

■«*<C0iN.-iOO00t-50i0
cOcOCDOcOiCiOUi'OiO
OOOOQOXOOOOQOOOQOOO

40 CONSTRUCTION OF MORTALITY TABLES

but the arithmetical work followed the equation

E[x]+n-i = Eix]+n + {e + w -\- d — s)[x]+«.

In these equations d(x]+n refers to the deaths in the nth insurance
year, and not to the deaths at curtate duration n.

Mr. W. J. H. Whittall (J. I. A., Volume XXXI, 167) intro-
duced a convenient notation to show in any formula for the
exposed to risk what methods were used in determining the age
at entry and the duration. Thus

X = Exact age
{x} = Age next birthday
(x) = Nearest age
I a; I = Mean age
t = Exact duration
t\ = Curtate duration
(t) = Nearest duration
\t\ = Mean duration

To the age symbols might be added ^j to indicate the com-
pleted age, or age last birthday, so that {x] = x\ +1.

This notation is appropriate for aggregate exposures, and for
calendar year and life year as well as policy year investigations,
but a difliculty arises in applying the age symbols to select
notation where the age at entry is indicated by [x]. This was
recognized by the author who stated

"It will be understood that in dealing with, for instance, age at exit {a;}
+ |f|, each symbol is variable, and the expression includes entrants of dif-
ferent ages and assured of different durations, subject only to the total ages
attained being correct according to the tenns of the formula."

Even in select notation, however, the duration symbols may be
used.

B. Calendar Year Method.

The calendar year method, as previously stated, is not directly
applicable to the construction of select tables. In considering
the formula for the exposed to risk in an aggregate table, it is
essential to define the symbols carefully.

Let Ux = New entrants aged x on January 1st of the year of
entry.
Cx = Existing at age x on December 31st of the last year
covered by the investigation.

FROM THE RECORDS OF INSURED LIVES. 41

Wx = Withdrawals in calendar year beginning at age x.
6x = Deaths in calendar year beginning at age x.
<Tx = Entering observations at age x on January 1st of the
first year covered by the investigation.

Let it further be assumed that an investigation of the data
justifies the use of the fraction / as the part of a calendar year
elapsed at the average date of entry, and the fraction g as the
average part of a calendar year elapsed at the date of withdrawal.

Then

X XX X x — \

Ex = Hnx - frix + ^<Tx - ^Wx + gwx - ^Bx - ^dx,

0 0 0 0 0

Ex+1 = Ex-\- rix+i — /(wj+1 — nx) + Cx+i — Wx+i

+ giwx+i — Wx) — ex+1 —

(4)

It is usual to assume that / = gf = |, in which case this equa-
tion is reduced to

F - JP M ^»+l + '^' M rr «^»+l + Wx ,.>.

iLx+i = iix T 2 ^ *^*+i 2 "^"^ ~ ^

If a truncated table were to be constructed, eliminating the
experience of an integral number of years, t, from issue, the same
equations would apply; but the age at entry would he x -{- t
instead of x and all policies terminated for any cause at the end
of t years or before would be eliminated from the experience.

When the age at entry is taken as the nearest age or mean
age at issue, the exposures and deaths are given for half ages
instead of integral ages as mentioned in Chapter III. Thus
Hx could be described as the new entrants aged a; + ^ at date of
issue, a; + i being integral. Equations (4) and (5) are still
applicable, but the ages for all the symbols must be taken as
half ages.

In forming the H^ table the exposed to risk were calculated
in the following manner :

Ex = Ex-i + U^x + rix+i) — liwx + Wx+i) — (dx + ex).

Where* nx = Entrants at age x next birthday.

* As the experience covered all policies from their dates of issue, there
were no cases entering observation after issue.

CONSTRUCTION OP MORTALITY TABLES

Wx = Withdrawals at age x next
birthday

dx = Deaths at age x next birth-
day

e, = Existing at age x

i.e., age at
entry next
birthday plus
mean dura-
tion = age
at end of cal-
endar year of
exit

This formula assumes that the observations close at the end of
a calendar year.

In investigating the data of the Scottish offices which con-
tributed to the Institute Experience, Mr. Meikle transformed
this formula as follows:

i,nx - i.{dx + Wj: + ex) + K^x-i-i - 'Wxfi)

= Zw, - E/x + M^x+i - W^x+l)

where /x = dx + Wx + e^. Mr. Ryan (J. 7. A., XXVI, 257)
points out that Mr. Meikle's form is easier to apply than the
Institute form, but suggests a further modification to the follow-
ing:

Ex = Z(Wx - /x) + K^x+l - Wx+l)'
0

This would save one step in the calculations.

It should also be kept in mind that the data entering into this
experience were separated by age at entry, in order that the
effects of selection might be seen and approximate select tables
prepared at a later date. The following illustration will show
the arrangement of the figures :

* Current Age at Entry 30.

Number of Entrants 6791.

Current Age
at Exit.

Existing.

Discontinued.

Died.

30
31
32
33

319
252
230
235

75
365
220
153

4
28
35
49

*Current Age means age next birthday.

FROM THE RECORDS OP INSURED LIVES. 43

The calendar year of entry, called "Year 0," gives the exposed
to risk for age 29. The next calendar year, or "Year 1" refers
to age 30 and so on. Thus by the Institute formula for this one
age at entry

£729 =^-y= 2858,

E^o = ^29+ ^ - Z5_+365 _^^_^ 3jg^ ^ ^210.5.

In forming the H™ table the figures were combined for each
attained age, irrespective of age at entry.

C. Life Year Method.

As the life year method is not suitable for the construction of
select tables, consideration will be given only to aggregate
tables. Let us suppose that exact ages at entrance and with-
drawal have been tabulated.

Then let rix = New entrants at age x last birthday.
Wx = Withdrawals at age x last birthday.
6x = Deaths at age x last birthday.

Let it also be assumed that the aggregate of the fractions of
a year since last birthday at the time of entry of n^ are tabulated
and denoted by /«, and that Qx is the sum of the fractions of
exposures in the year of withdrawal among Wx.

Then

X X x—l

Ex = T,nx - fx - T.Wx -\- 9x - Z^x ra\

0 0 0 \p)

Ex+1 = Ex + rix+i — (fx+i - fx) - Wx+i + (gx+i - gx) - Qx-

If instead of using fx and gr,, which are the sums of the fractions
and vary with the attained age, average fractions per entrant and
withdrawal, / and g, are adopted for all ages, we have

X X x—l

Ex = T.nx - fux - T.Wx + gwx - Z^x,

0 0 0

Ex+1 = Ex + rix+i — fiux+i — rix) — Wx+i

+ giwx+i - Wx) - dx
= Ex+ {fux + (1 - /)n^i}

- {gwx-i- {I - g)wx+i} - Ox.

If there are persons under observation when the period of
exposure began and existing at the end of that period, the former

(7)

44 CONSTRUCTION OF MORTALITY TABLES

may be treated as new entrants at their ages at which they came
into the experience and the latter in the same manner as with-
drawals. In such a case they must of course be taken into
account in arriving at the average fractions / and g.

It has been stated in Chapter III that the ages at entry and
withdrawal may be taken as the mean ages. It might be con-
tended that we then have a calendar year method. This is not
the case, however, as under the latter method the deaths are
analyzed by calendar years, i.e., by age at the beginning of the
calendar year of death. Under the life year method the deaths
are analyzed by the age last birthday at death, and mean ages
at entry and withdrawal are used simply as a device for getting
rid of fractional exposures at such times.*

If the observations begin and close with persons under ob-
servation or existing, their ages at such times could not or-
dinarily be taken as mean ages. The age nearest birthday
might be used. When, however, such exposures commence
and cease at the end of two different calendar years, another
convenient method may be adopted. Let <Tx represent those
entering on January 1st at the start, at mean age x in the pre-
ceding calendar year. They will be exposed for half a year
at age x as we may assume that on the average they are aged
X + I on January 1st, when the exposures begin. Similarly
Bx represents the existing on the final December 31st, age x
being the mean age in the year just ended. There will there-
fore be on the average a half year of exposure at age x. Then
if X represents the mean age in the symbols Ux and Wx, while 6x
represents the deaths at age x last birthday, we have

X X X — 1 X X

Ex = Z^x + Z<^s — h<^x - Z^x - T.^x — T.^x + h^x,

0 0 u 0 0

E^+1 = Ex^- Wx+1 + hicx + cx+i) - dx (8)

- Wx+1 - Kcx + ex+i).

If the period of investigation had not extended over exact
calendar years, equation (8) could still be used if x in the symbols
(Tx and Cx were the age last birthday, for we should still have
approximately one half year of exposure at age x in both cases.

* Mr. W. J. H. Whittall (J. 7. A., Vol. XXXI, 163) discusses the definition of
methods and states (p. 164): "It would, therefore, seem probable that in the
system of determining the ages at death in any particular formula will be
foimd the key to its proper classification."

FROM THE RECORDS OF INSURED LIVES. 45

The last method of estimating exposures mentioned in Chapter
III is to take the ages at entry and exit as the ages nearest
birthday, except for the age at death. This produces no frac-
tional exposures, as those under observation at the start or
existing at the close of the period may be likewise treated and
included in the new entrants and withdrawals respectively.

Then

X r— 1 X

0 0 0 \^)

Ex+1 = -EJx + n^+i - dx - W:,+i.

One example of the life year method is the experience of the
Economic Office, 1862, taken out by J. J. Downes, who was the
introducer of the card system. A description of the methods
employed is given by W. J. H. Whittall, J. L A., Vol. XXXI, 171.
Exact ages to four decimal places were computed at entrance and
exit, which is an unnecessary refinement. The method of
tabulation was also laborious.

Mr. A. J. Finlaison used the life year method in compiling the
Government Annuitants' Experience, 1883. After a test he
assumed that on the average, four months had elapsed at date
of entry since the last birthday. Accordingly each new entrant
was assumed to be exposed to risk for two-thirds of a year at
the age last birthday at entry. Ages at exit were taken ac-
curately, the exposures on the existing terminating on exact
birthdays. Mr. Finlaison used the data to form select tables of
annuitants' mortality by assuming that the eight months mor-
tality rate following entry was the rate for the first year after
entry, and so on. Thus the approximate periods of experience
0-f, f-lf, 11-21, etc., were considered to be annuity
years 1, 2, 3, etc. This illustrates how, in attempting to get
mortality rates for each exact year of life, the advantages of
accurate tables by policy years must be sacrificed. As the
effect of self-selection by annuitants is often marked for the
first few policy years, the understatement of the mortality in
the early policy years under Mr. Finlaison's method would
appear to be more important than the loss of accuracy that would,
be introduced by the use of the policy year method. The
annuity experience of the British Offices, 1863-1893, was based
on annuity years, the age at entry being the age nearest birthday
at date of purchase.

Chapter VI. Tables with More than One Decremental

Factor.

A mortality table, as heretofore considered, is a table showing
the number living at each age, either in the select or in the aggre-
gate form. This number decreases as the age increases only
because of the deaths. Death is the only decremental factor
in such a table. We may, however, have a table which has one
or more additional decremental factors, such as those of with-
drawal, marriage or the re-marriage of widows. If the addi-
tional factor is that of withdrawal, then the table will show for
each age the number living who have not withdrawn, and in
proceeding from one age to the next there is deducted the number
who have withdrawn and those who have died without with-
drawing during that year of age. There is given below a speci-
men table (J. /. A., Vol. XXXIII, 196) in the select form for
age 20 at entry:

Mortality Table,

Showing Mortality and Withdrawal.

Deaths

Dura-

Numbers

Deaths.

With-

and

tion.

Living.

drawals.

With-
drawals.

(0.

^M+t-

<^W+<-

WM+t.

(d +u>)(xi+i.

100,000

426

14.970

15,396

84.604

373

10,549

10,922

73,682

335

7,717

8,052

65,630

307

5,892

6.199

59,431

297

4,741

5,038

Before starting to analyze the experience from which it is planned
to construct such a table we must examine the functions by
which we may compute the successive figures in the table. The
rate of withdrawal will ordinarily vary with the duration of
insurance as well as with age and therefore select tables are
best adapted to display the results. It may also be noted that
the rate usually varies with the plan of insurance, making sep-
arate tables by plan desirable in many instances. The investi-
gation might be based on lives, policies, or amounts insured.

FROM THE RECORDS OF INSURED LIVES. 47

Let US adopt, the following notation:
Table Notation:

l'ix]+t = the survivors at age x -}- t of the entrants at age

X who have not withdrawn,
d'ix]+t = the deaths during the (t + l)th year, before

withdrawal.
w'[x]+t — the withdrawals during the (t + l)th year.

Investigation Notation:

Eix]+t = the exposed to risk of death.
(wE)ix]+t = the exposed to risk of withdrawal, the deaths
being given their respective fractional
durations while the withdrawals are given
a full year's exposure.
Oix]+t = the deaths in the {t + l)th year.
W[a!]+t = the withdrawals in the {t + l)th year.

It may be noticed that the definition of (wE)[x]+t is consistent
with the method by which Eix]+i is determined, as in the latter
case the withdrawals are given their proper fractional exposures
but the deaths are counted as exposed to risk for the entire year
of death. Thus to find the proportion that withdraw of those
persons who are exposed to withdrawal for a year, we must
count each person who withdraws as a unit and not a fraction
in the exposures.

In tabulating W[x]+t the question arises whether the with-
drawals at the exact end of a policy year, which are generally
a large proportion of the total withdrawals, should be grouped
with those during the year just closed or with those of the sub-
sequent year. If we consider the first policy year it will be seen
that the former course must be adopted in order to get a full
year of withdrawals recorded in that year. We shall then have
the rates of withdrawal during and at the end of each year, and
l'[x]+t will represent the survivors of the original entrants who
continue their policies beyond t years from issue. It should be
remembered, however, that not all the l'ix]+t persons continue
their policies to the end of the {t + l)th year, for some will
withdraw at fractional durations.

If we now consider the double decrement table we see that the

48 CONSTRUCTION OF MORTALITY TABLES

probability that a life aged [x] + t will withdraw within a year is

which is different from the rate of withdrawal, among those
who are alive. This latter rate is

i\x]+t—hd'ix]+t

on the assumption of a uniform distribution* of deaths, and is

the rate given in an investigation by the ratio W[x]+il(wE)[x]+f

Similarly the probability of a life dying before withdrawal

becomes

^f _ d'ix]+t
Q w+t - 77 >

whereas the rate of mortality, on the assumption of a uniform
distribution of withdrawals, is

Qlx]+t — 77 i:;^ •

The value of q[x]+t is given in the investigation by dix]+t/E[x]+t'
It is evident that in a single decrement mortality table q' becomes
g, because the column of survivors, I, is reduced only by deaths.

In order to construct a table from the data we must find rela-
tions by which, starting with a given radix, we may find the
values of d'lxut and w'[x]+t- We can then construct the V column
by the equation I'm+i+i = l\x]+t — d\x]+t — «>'[*]+<.

The simplest procedure would appear to be to calculate the
values of q' and (wq) directly from the data, for these prob-
abilities when multiplied by I' give us d' and w\ If this were
done we should find the probabilities from the data as follows.

q w+* = p7 — > UU;

{wq),xu^=j^^^^, (11)

where E' and {wE') are obtained by giving 6[x\+t and tftii+t a

* It should be noticed that this assumption as applied to a double decrement
table is not equivalent to the assumption of a uniform distribution of deaths
in a single decrement table.

FROM THE RECORDS OF INSURED LIVES. 49

full year's exposure in both cases, and therefore

E\x]+t = {wE')[x]+f

Using the notation for single decrement tables except for the
change in the definition of W[x]+t, and assuming that the period of
observation begins and ends on policy anniversaries, we have

t <— 1 t t—\ t

0 0 0 0 1

where T[x]+t represents the matured, such as endowments and
terms, taken to mean or nearest durations. Then

E\x]+t+i = E'[x]+t + o'[x]+<+i

— i'^ix]+t + G[x]+t + T[x]+t+i + e[x]+t+i)' (12)

This gives us the denominator common to equations (10) and (11).

It frequently happens that the probabilities q' and (wq) are not
as useful as the rates q and (wr). If we wish to compare the
results of the investigation with those of other experiences the
latter rates should be used as q is independent of (wr) and vice
versa, while q' and (wq) are each functions of both the mortality
and the withdrawal rates. It may be desired to combine the
withdrawal rate from one experience with the mortality rate of
another, in which event we need to have the independent rates,
q and (wr).

Let us then see how we can construct the double decrement
table from these rates.

On the assumption of a uniform distribution of deaths and
withdrawals during policy years we have

(V - W)q = d\
(V - hd')(im') = w'.

Solving the equations we have

^, ^l'q[l -h(wr)\ ^ (13)

^ V(wr)[l - M . (14)

1 - \q(wr) ^ ^

We may thus compute the decrements from the rates q and
(wr), which can be obtained from the deaths, withdrawals, and
exposed to risk as previously shown. We can also construct
a table where q is derived from one experience and (wr) from

60 CONSTRUCTION OP MORTALITY TABLES

another. We may also obtain a check formula for the V column,
as follows:

l+i' = 1' -d' -w'

^ „ r _ q-h (wr) - q{wr) 1

*L \-\q{vyr) J (15)

^ ^,(1 -g)[l - {wr)]-\q{wr)
1 - \q{wr)

These equations were developed by Dr. T. B. Sprague in a
paper on marriage and mortality tables (J. /. A., Vol. XXI, 406)
and were applied to tables of mortality and withdrawal by Mr.
T. G. Ackland (J. /. A., Vol. XXXIII, 194). Mr. Ackland, in
his paper, discusses very fully the use of exact, mean and nearest
durations in obtaining the exposed to risk, E[x\^t and {wE)ix\Jrti
and concludes that the nearest duration method is, upon the
whole, the best suited for obtaining the rates of mortality and
withdrawal from a large body of insured lives.

Mr. Ackland deduces the formulae for the exposed to risk
where the period of investigation extends over integral policy
years in the following manner.

Let S[ai+t = the survivors in force at the commencement
of the period of observation, at integral
duration t.
n[x\ = the new entrants at age x.
^[x]+t = the cases existing at the close of the period of
observation, at integral duration t.
{aw)[x\+t = the withdrawals having a duration greater
than t that are treated as having integral
duration t, i.e., those with a duration less
than < + .5 and one-half those having a
duration of exactly t + .5.
(bw)[x]+t = the balance of the withdrawals with duration
greater than t but not in excess of < + 1.
These are treated as having integral dura-
tion t -\- 1.
Then {aw)ix]+t + (hw)ix]+t-i = Ww+t, the total
withdrawals at integral duration t by the
nearest duration method.
{ad)ix]+t = deaths in first half of {t + l)th year,

FROM THE RECORDS OP INSURED LIVES. 61

(hd)ix]+t = deaths in last half of (t + l)th year, and
letting d[x]+t = {ad)[x]+i + (6d)[*]+<-i*

also, let

where
Then

(e + M) 4- d)M+t = Fir]+t
(s — F)[x]+t = G[x]+t

t = 0, G[x] = nix] — Fix].

Eix]+t = iiG) + (ad)ix]+t (16)

(wE)ix]+t = JliG) + {aw)ix]+t (17)

On page 52 is given a copy of the working sheet used to carry
out the tabulation for age 20 at entry.

Mr. Ackland's notation is retained in this table. It must be
carefully noted that W[x]+t in columns (4) and (13) are not the
same functions, and a like condition is true of dix]+t shown in
columns (5) and (10). Column (11) is obtained by dividing
(10) by (9), and column (14) is the quotient of (13) -4- (12).
The brackets in column (5) indicate how the numbers in columns
(4) and (5) were combined to obtain (6). Columns (10) and
(13) were obtained from (5) and (4) respectively by combining
the figures set down opposite each duration.

The data being analyzed had been obtained from a Clerks'
Association in which subscriptions were payable monthly on
the first day of each month. Therefore the withdrawals could
be assumed to be distributed uniformly over each year of dura-
tion. Mr. Ackland found that the exact average fractional
duration at withdrawal was approximately one half year. Ac-
cordingly formulae (13), (14) and (15) can be considered suffi-
ciently accurate.

Dr. Sprague, in the paper previously mentioned, also developed
for a double decrement table central rates corresponding to

^ 2 2

* It will be seen that rf[i]+« has not the usual significance of 6ix]+t or d[x]+t
but is calculated similarly to wix]+t in accordance with nearest durations.

CONSTRUCTION OF MORTALITY TABLES

S d

^_^

3.2

■^

I2<=>

•*

Q -JS

"-'

1^1

<-(

t«.

I r

> •-<

i-i

1-1

-a

:^^l
^•s^

eo lO

^eo

""f

■<J<

■*

? § f

^«

»H

1 " +

T}<

+'«|KJ

<-!

a »

s .

I-I

+ § 4

05 ^

fH
CO

rt.

Oeo

'*3 II +

r-v"-!

00 <-'

1-4

»o

WOO

IfcJ

Net
Move-
ment.

^ II

o
1

'S- i

Total
Decre-
ment.

+ +

^-K<-<

ffl (N

1-1

-I*

o
1-1

i + i

1 «

i TT i

TT .5. TT

O CO

(M O

'HO

(NO

IS"

J« "-• •*

CO t*

o»t^

s;2

00 -^

TftOO

(N <-^

1.^ fH

(N .-I

«:l

s •

s '

C???

d.^

^00

1-1

3.2

3°

T-*

•*

»o

FROM THE RECORDS OF INSURED LIVES. 53

and

(wm) =

w
* 2 2

which are mutually independent but from which can be computed,
if desired, the rates q and (wr) by the relations

3 =

2 + m

and

/,„.^ _ 2(w;m)
(_wr) =

2 + (wm)

Central rates were employed by Mr. Ackland, who shows that

and

and also that

d' = V

w' =V

m + (wm)
"^ 2

(«jm)

^ w + (i«wi)

1 -

w + (rum)

1 w + (ww)

In computing the central rates he divided 6[x]+t and W[x]+« by
the adjusted exposed to risk, E[x]+t, in each case. Eix]+t was the
number exposed to risk computed up to the actual cessation of
the risk from any cause — under the nearest duration method,
to the nearest integral year. This makes it unnecessary to
calculate separate exposed to risk for deaths and withdrawals.
These equations are, however, incorrect where there is not a
uniform distribution of withdrawals.

A life insurance company which issues policies only with
annual, semi-annual and quarterly premiums could not assume
a uniform distribution of withdrawals. For example, assuming
that lapse is as likely at the second semi-annual premium as at
the first, the average fractional duration would be

Ks*'

)-l

64 CONSTRUCTION OF MORTALITY TABLES

and similarly for quarterly lapses

4\4^2^4^ / 8

If we assume 62^ per cent, of the lapses are on policies issued at
annual rates, 32^ per cent, at semi-annual, and 5 per cent, at
quarterly, the average fractional duration for all lapses would be

.625 + .325 X 7 + .05 X i = .9.
4 8

If we felt warranted in considering that this represented the
proper average fraction for all withdrawals, including surrenders,
at all durations, we might record W[x]+t in column (4) as equal
to .9wix]+t-i + .lw[x]+t, i.e., refer nine-tenths of the withdrawals
during or at the end of any year to the integral duration at the
end of the year, and one-tenth to the beginning of the year,
instead of tabulating by the nearest duration method the func-
tions (aw) and (bw).

This example points out the error in assuming in such a case
that

{l'-l«')i-d'.

Continuing the above illustration it would be necessary to
change this equation to

{V - .lw')q = d',

which, together with

(r-id')(«;r) =w',

might be solved to give d' and w' in terms oil', q and {wr).

In the discussion of Mr. Ackland's paper, Mr. R. Todhunter
presented a new view of v/ithdrawal rates. He states :

"I suggest, in the first place, that discontinuances in ordinary business are
attributable to a force essentially different in character from the force which
operates to produce mortality. Death claims are caused by a force which
operates continuously throughout the history of policies, whereas discon-
tinuances are caused — so far as regards the large majority of policies — ^by
a discontinuous force coming into operation at certain recurring epochs. A
pohcyholder does not exercise his option of withdrawing continuously in the
same sense that he is continuously subject to the risk of death; he exercises it
periodically, in most cases on the occasions of his receiving a renewal notice,

FROM THE RECORDS OF INSURED LIVES.

and having to decide whether to pay or not to pay. This applies, I think,
to nearly all lapses, the only exceptions that occur to me being the few cases
in which a poUcyholder pays a series of fines to extend the days of grace for
successive short periods; and lapses constitute a very large proportion of
the whole number of discontinuances. It applies, also, to all surrenders
carried out at or about the renewal date. The proportion which these form
of the entire body of surrenders, will no doubt vary in different classes of
business; in a small experience of two years, I found that 55 out of a total of
94 surrenders took place during the days of grace, 11 took place within a
month before the renewal date (probably on receipt of renewal notices), and
the remaining 28 were scattered. Having regard to the small proportion
that the number of scattered surrenders forms of the entire number of lapses
and surrenders, I think it may fairly be stated as a general proposition, that
discontinuances are mainly due to the exercise of a periodical option at or
about the renewal date.

"If this proposition be admitted, it follows that the force which causes
discontinuances would be more appropriately measured by rates of non-
renewal, than by rates or forces of withdrawal — in other words, by the ratios
that the withdrawals at definite epochs bear to the exposed to risk of with-
drawal at those epochs, than by the ratios that the withdrawals in given
periods bear to certain numbers supposed to be continuously exposed to the
risk of withdrawal throughout those periods. As applied to a collected ex-
perience, this second proposition presupposes a pohcy-year tabulation of the
observed facts. A tabulation by calendar-years, or years of life, will, of
course, have the effect of spreading the discontinuances over the years of
observation, and will thus exhibit something of the nature of a continuous
force of discontinuance. If it be admitted that such a force has no real ex-
istence, the fact that it is artificially created by any method of tabulation
other than one that follows the years of assurance may be considered another
argument in favour of the PoUcy-year Method."

The sort of table suggested can be clearly seen from that
given below. *

Year

Exposed to

Rate of

Risk

Q-

Risk of Non-

Non-

Aasurance.

of Death.

Renewal.

8,016

.0037

7,986

1,288

.161

6,313

.0057

6,277

476

.076

5,535

.0045

5,510

297

.054

4,945

.0059

4,916

149

.030

4,404

.0057

4,379

176

.040

3,958

.0038

3.943

.025

Total

33,171

160

.0048

33,011

2,481

.075

The nearest duration method is a convenient one for arranging
the withdrawal data in the proper form for determining the rate

* Terminations other than by death and withdrawal are not shown in this
table.

56 CONSTRUCTION OF MORTALITY TABLES

of non-renewal. Each of the numerous withdrawals at the
end of the policy year would be located exactly. Those at
semi-anniversaries would be alternately located at the beginning
and end of the year, lapses at the end of the first and third
quarters would be thrown to the nearest anniversary as would
also the usually small proportion of surrenders that are scattered
through the year. The result would be that all withdrawals
would be considered as taking place on policy anniversaries.
This distorts the facts somewhat, but on the whole gives a
more nearly correct view than to assume that all the withdrawals
are spread throughout the policy year. Mr. Todhunter sug-
gests that it may be desirable in a large experience to exhibit
the rates of mortality and non-renewal by quarters of a year
for the first two policy years.

It may also be seen that a double decrement table constructed
in this manner is a useful one for determining the present value
of future commissions, because V[x]+t will be the number of
survivors at duration t who pay premiums to the end of the
(t 4- l)th year. The table could also be used for comparative
purposes and adjusting premium rates to provide for varying
rates of non-renewal (J. I. A., Vol. XXXIII, 277, 278).

In obtaining the exposed to risk and rates of mortality and
non-renewal we may in general record the data as shown in
Chapter V by the policy year and nearest duration methods for
a single decrement table, but we must separate the matured
from the withdrawals. Furthermore we must include in <Tix]+t
all policies terminated prior to the period of investigation at
nearest duration t and those terminated at exact duration t.
These additional policies will be included also in W[x]+t so that
they will both enter and also withdraw at duration t. This
will have no effect on the exposed to risk of death but will give
the necessary data for the rate of non-renewal at duration t.
If <r[x]+t were not to include these cases, such entrants would
have to be excluded from the exposed to risk of non-renewal
at duration t, making it necessary also to exclude the withdrawals
at durations between t and t -\- .5 among such entrants. This
could be done by excluding from o-[x]+t all cases terminated by
withdrawal at durations less than t -\- .5 and half of the with-
drawals at that exact point — as this would not affect Eix]+t — and
by excluding <Tix]+t from the exposed to risk of non-renewal at
duration t. Let us assume that the former course has been
adopted in order to give additional data.

FROM THE RECORDS OF INSURED LIVES. 57

Similarly for the existing, let us suppose that eix]+t excludes
policies withdrawn at nearest duration t even though they may
have withdrawn after the policy anniversary, and that the
policies so excluded are included in W[x]+«- Then eix]+t will be
included in the exposed to risk of non-renewal at duration t.

If we represent the matured at nearest duration t by T[xutj
formula (3) for the exposed to risk at death will become

E[x]+i+i = E[x]+i + <rix]+t+i

— C^lxl+t+l + Wix]+t+l + ^M+t+l + d[x]+t) (18)

and

Qlx]+t =

[*]+<

the rate of mortality and also the probability of death in the
double decrement table. Also (nrE)[x]+t, the exposed to risk
of non-renewal at the end of the (t)th year,
t t

t-i t-i <— 1

— zl "^M+t — zL ^w+« — zl ^i']+t-

Then

(nrE) [x]+t+i = inrE)[x]+t + <Tix]+t+i

— (T[x]+t+l + W[x]+t + €[x]+t + d[x]+t)
= Eix] + t + <Tlx]+t+l — T[x] + t+l — dix] + t'

If then we let

fix]+t = wix]+t + eix]+t
and

9M+t — <^{x\+t — T[x\j^t — ^[i]+<-i
we have

E[x\Jrt+i = Eix]+t + gix]+i+i — fix]+t+i (19)

and

(nrE)[x]+t+i = E[x]+t + gM+t+i- (20)

The rate of non-renewal at the end of the (<)th year is

Wlx]+t

(nrq)

lx]+t —

(nrE)

lx]+t

Having calculated the rates qix]+t and {nrq)ix]+t we can con-
struct the double decrement table by the relation

l'lx]+t+i = (l'ix]+t — q[x]+t l'[x]+i)[l — (nrq)ix]+t+i]

= l\x]+t(l - qix]+t)[l - (nrq)[x]+t+i]. (21)

58 CONSTRUCTION OF MORTALITY TABLES

This form of the equation shows that the probabilities of living
and of renewal are entirely independent of each other.

The data which are used as the basis of the rates of mortality
and withdrawal may include policies issued with deferred dividend
periods. Such policies were so often sold with emphasis on the
total cash surrender value available at the end of the dividend
period, that it may be considered advisable to treat them as
matured on the dividend due-date.

Another point which may be mentioned is that it may be
found, in any particular experience, that the non-renewal or
withdrawal rate may vary with duration only for a certain
number of years. The rate might even become practically
constant for all attained ages beyond a fixed duration. In
either case tables of mortality and withdrawal could be cal-
culated in the select and ultimate form, the duration of the
select period being the greater of the select period of mortality
and the select period of withdrawal.

Tables of mortality and withdrawal have been the commonest
forms of double decrement tables based on the records of in-
sured lives in America, but in recent years tables having de-
crements from mortality and total and permanent disability
have assumed great importance. The probabilities involved
together with formulae for the exposed to risk are given in
Actuarial Studies No. 5.

In Great Britain the subject of marriage and mortality has
been of importance to actuaries because of such contingencies
that have been occasionally insured against by the British
Companies. Dr. Sprague's paper presented in 1879 {J. I. A.,
Vol. XXI, 406) has already been mentioned. His formulae,
developed on the assumption of a uniform distribution of deaths
and marriages throughout the year, are applicable to that sub-
ject, though introducing a considerable error when used for
withdrawals in ordinary insurance experience. In a later paper
(J. /. A., Vol. XXVIII, 350) Dr. Sprague has given tables for
the re-marriage of widowers, showing the necessity for a select
period for "recent widowers" merging into an ultimate, or
"chronic widowers" rate at the end of twenty years for those
who become widowers at age 25, the select period shortening
as the age increases. In a paper in the same volume (p. 384)
Mr. J. Chatham brings out that in constructing the select
portion of such tables we may obtain ^{wl)^x\+t, the widowers,

FROM THE RECORDS OF INSURED LIVES. 69

becoming such at age x, who are alive and have not re-married
at age x ■\- t, by working backward from (m>Z)[xi+«+i by the
relation

where m and m are respectively the central marriage and death
rates at age [x] + t, and similarly for the individual decrements :

(«?m)[xi+t, widowers marrying, = {wl)[x\+t+\- ^ _ ir — X — ]
{wd)[^]+t, widowers dying, = (w;Z)[xi+«+i- ^ _ rr^^ ^ \ '

These equations are obtained by transforming the equations
corresponding to those contained on p. 53.

Messrs. Hewat and Chatham investigated the mortality and
marriage experience of the Widows' Funds of the Scottish Banks
(J. /. A., Vol. XXXI, 428) by calculating the central rates.
The period of investigation was closed for the different banks
in such a way as to lead to the assumption that the existing were
exposed to risk on the average for six months after the age at
which they were recorded as terminating. They describe their
method as being analogous to the exact duration method.
Using (6m) J to indicate the number of bachelors marrying be-
tween the ages x and x -\- 1 and {hd)x to indicate the bachelors
dying within that year of age, they employed the following
formula for the number of bachelors exposed to risk of both
marriage and death at central age x + I-

Ex+H = 2(n;,_i - /x_i) 4- Wx - \[ex -\- Wj. -\- (bm)x + {bd)^]
= S(nx_i — fx-i) 4- rix — Ifx,
where

fz = Cx -{- Wx -\- {bm)x + ibd)x.

It is important to note the simplicity of this method which re-
quires no separation of both (brnjx and (bd)x into two parts
similar to Mr. Ackland's method of treating withdrawals and
deaths. In the above formula it is evident that the withdrawals,
Wx, were recorded as terminating at their curtate durations, as
well as those marrying, dying, and existing. The accuracy of
assuming one half a year of exposure beyond curtate durations
for all terminations should of course be tested before the assump-
tion is made.

60 CONSTRUCTION OF MORTALITY TABLES.

Mortality tables based upon the records of insured lives which
contain more than two decremental factors would rarely be
constructed. The principles involved are the same as those
covered by the preceding discussion of double decrement tables.
The computation of central rates will usually be found of ad-
vantage, because the exposed to risk has the same value for
all central rates at the same age, and from the central rates the
table may be constructed and also the annual rates (per year of
exposure) may be calculated. These latter rates will often be
wanted in order that they may be compared with similar rates
from other experiences. It might also happen that some of
the rates entering into the multiple decrement table are to be
obtained from the investigation while others are to be taken
from a different experience. In such a case we must obtain the
central or annual rates, because we cannot compute the prob-
ability of a life surviving and otherwise remaining in the re-
quired status unless all the contingencies, as for example, death,
total and permanent disability and withdrawal, entered into
the single investigation.

Chapter VII. Miscellaneous.

There will now be considered various points in connection with
the construction of mortality tables which have not been pre-
viously discussed.

Final Series Method: In compiling the mortality experience of a
life insurance company, it will be found that while there may be
many old policies which have run their course, there will be a
considerable number existing at the close of the observations.
In the preparation of the mortality experience of the Thirty
American Life Offices, which was based on calendar years and
is described in Actuarial Studies No. 1, it was found that the
average duration of the policies was only 4.36 years, and that of
982,734 male lives entering into the investigation 527,157 were
existing at the close of the observations. The method of Final
Series was adopted to carry the existing forward to their ultimate
destination of death or withdrawal, and this was done on the
assumption that the future experience on the existing would
follow the same select rates of mortality and withdrawal that
had actually been experienced in the past. Incorporating this
hypothetical data obviously has an effect only on aggregate
tables. The method employed may best be illustrated by an
example taken from page 36 of the volume* giving the results of
the investigation. The table on page 62 for one age at entry
represents amounts to the nearest thousand with 000 omitted.

Although the values given represent thousands of dollars of
insurance it will be easier to discuss them as though they repre-
sented lives or policies.

The original data show that there are 11,763 existing at the
end of year 0 (i.e., the first calendar year of insurance) to be
carried forward, and out of 176,799 entrants of year 1 there are
27,361 discontinued and 1743 in claims in that year. In the
final series the entrants on year 1 include the original 176,799
entrants and the 11,763 existing at the end of the previous year,
making a total of 188,562. It is assumed that the discontinued
and claims arising out of the 188,582 will be in proportion to the
discontinued and claims arising out of the 176,799 entrants in

* System and Tables of Life Insurance — Meech, Vol. I.

CONSTRUCTION OF MORTALITY TABLES

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FROM THE RECORDS OF INSURED LIVES. 63

the observed data. The discontinued and claims in the final
series will therefore be found by multiplying the amount of
these items in the original data by 188,562/176,799, namely, by
1.0665. This gives 29,181 discontinued and 1,859 in claims,
a total of 31,040; which subtracted from 188,562 gives 157,522,
the entrants on year 2. It may be pointed out that the number
of entrants on year 0 is the total of those passing out of observa-
tion as the existing, discontinued and claims, namely, 102,484
+ 84,429 + 9,549, or 196,462; and if the adjusted sum of dis-
continued and claims is deducted each year the balance gives
the entrants on the following year. In this way the total ex-
isting according to the original data is automatically taken care
of in the final series by merely calculating the amounts passing
out of observation by discontinuance and claims.

The exposed to risk of death after year 0 is obtained by sub-
tracting from the entrants one half the discontinued. For the
year 0 the exposed is made up of half the entrants, less half the
discontinued, or what is the same thing, one-half the difference
between the entrants and discontinued for that year. The
amount of claims divided by the exposed to risk of death gives
the qx values.

The exposed to risk and deaths for the first calendar year are
unaffected; for the second year they are increased by a small
percentage, for the third year by a larger percentage, and so on
till the twenty-eighth year when the process terminated. The
process as illustrated in the foregoing schedule resulted in a
multiplier greater than 100 for some ages at the longest durations,
a figure which appears to have alarmed the computers, as the
multipliers finally adopted were those brought out by the
original process multiplied by (1.04)~", where n represents the
number of the year of insurance. Even with this adjustment
the data of the later durations would at some ages be multiplied
twenty and thirty fold. The hypothetical data for the aggregate
table thus included an enormously larger proportion of experi-
ence at the longer durations than did the actual material, re-
sulting in a considerable increase in the rate of mortality, especi-
ally at the older ages. It is open to question whether the
final series method may not disturb the mortality rates to such
an extent as to render the results untrustworthy. So far as we
know it has been applied only in connection with the 30 American
Offices' Experience.

64 CONSTRUCTION OF MORTALITY TABLES

King's Method: Mr. George King in Volume XXVII, page 218,
of the Journal of the Institute of Actuaries, described a method of
ascertaining the mortality experience of a life insurance company
between valuation dates. It is quite customary for British life
insurance companies to make a valuation of policy liabilities and
to distribute surplus once in five years. A yearly valuation is
sometimes made for the guidance of the companies, but the
valuation on which the quinquennial distribution of surplus is
based is made once in five years, and King's method was de-
signed for the purpose of readily ascertaining the mortality
experience of a company between valuation dates.

Assuming that the valuation date is December 31st, and
changing the notation used by Mr. King, the method is as
follows:

Let Sx = survivors at commencement of observations where x

is the age nearest birthday on December Slst,

the day before such commencement;
w, = entrants during period of observation, where x is

the age nearest birthday on nearest December Slst

to date of entry;
Wx = withdrawals, where x is age nearest birthday on

nearest December Slst to date of withdrawal;
6x = deaths, where x is age nearest birthday on December

Slst preceding death;
Bx = existing at December Slst at close of observations,

X being age nearest birthday at that date; and
Ex = exposed to risk in year of age x to x + 1.

Then:
and

Ex = Ex-i + Sx -\- Ux - Wx - 6x-l - €x

^' = K

The Sx and 6, are taken directly from the classification registers
which are used in connection with the valuation of the policy
liabilities and it is necessary to classify merely the n,, Wx and 6x-
Furthermore, fractional exposures are avoided. Although in-
tended to apply only for a short term of years, this method might
be extended to apply to the entire mortality experience of a
companj'.

FROM THE RECORDS OF INSURED LIVES. 65

Mr. King points out that, as compared with the Institute
Method, his formula avoids exposures of fractions of a year and
is more accurate where entries and exits are not uniformly dis-
tributed throughout the year. Moreover the Institute Method
gives no exposure to lives entering and discontinuing in the same
calendar year, while approximate allowance is made for them
under Mr. King's method. He calls attention also to the rapidity
of the calculation of the exposed to risk by summation, as follows:

Ex = S(sx -\- Ux — Wx — dx-i — ex)
and letting

Sx -\- Ux = hx
and

Wx + dx-i + ex = fx)
then

Ex = S^i — 2/j!
= ^{hx-fx).

Mr. King's method was intended to be applied in British Life
Offices where the method of valuation is different from that
followed in this country. The majority of the business was
tabulated according to the oflfice age and was in convenient form
to apply the method. If it were desired to make use of this
plan in an ofl&ce using the American form of valuation, the
necessary particulars would require additional calculation.

Mr. Whittall has contended (/. /. A., XXXI, 184) that Mr.
King's method is one of mean ages. Looking at the question
graphically,

Dec. 31 Dec. 31 Dec. 31

June 30 I June 30 I June 30 I

an entrant in the period he is relegated to 6 and his nearest
birthday at point 6 must lie between a and c; an entrant
in the period cd is relegated to d and his nearest birthday at the
point d must lie between c and e. This is equivalent to taking
mean ages with years running from June to June instead of
December to December. This is apparent if it be assumed that
the office years run from June 30th to June 30th. Then an
entrant in the period cd would be relegated to c and his nearest
birthday at that point would lie between 6 and d, that is, within
the calendar year. Similarly an entrant in the latter half of

66 CONSTRUCTION OF MORTALITY TABLES

the financial year, de, would be relegated to e and at that point
the nearest birthday would lie between d and /, that is, within
the calendar year. In the case of an office with a financial
year ending June 30th, the same results would be attained by
adopting a system of mean ages, i.e., subtracting year of birth
from year of entry. The same principle applies to the survivors,
withdrawals, deaths, and existing, allowing for the proper in-
cidence of each.

Mr. King, in discussing Mr. Whittall's paper, disagreed that
it is a method of mean ages, because that method, as ordinarily
applied, permits as much as a year's error in age in any particular
case, while his use of the nearest birthday on the nearest De-
cember 31st, never permitted an error on the assumed date of
entry or exit of more than six months. They both agreed that
it was difficult to classify the method as one purely of policy
or calendar years. Mr. King, however, had previously stated
(J. I. A., Vol. XXIX, 178) that "the averages were so taken that
they really got policy years. The average was so arranged as to
make the policy years harmonize with the financial years of the
company."

Compilation of Select Tables where Data have not been Traced on
Policy Year Basis: A good example of the formation of select
tables from a calendar year experience is the select tables formed
by Dr. Sprague from the data of the Mortality Experience of
the Institute of Actuaries (J. I. A., XXI, p. 229). The data
followed calendar years and the lives were taken at age next
birthday at entry. It was assumed that the lives entered on
the average in the middle of the calendar year and attained age
next birthday at the end of the year. The lives were therefore
aged X — ^[at the date of entry in the middle of the year, x being
age next birthday at entry.

The values of ^px, liPx, 2iPxy etc., were first calculated for
each age (x — 1) at entry, the original exposed to risk being
modified to give an exposure of 3 months to those withdrawing
in the calendar year of entry. Then, in order to obtain sufficient
data the^experience was taken in groups of five ages. To avoid
giving undue weight to any particular age, 50,000 entrants were
taken for age {x — 2|), 100,000 for ages (x — 1^), {x — ^),
(^ + Difjand (x -h 1^), and 50,000 for age (x -f- 2^), making
500,000 in each group. By taking 50,000 at the youngest and
oldest age in'^each^group an even 100,000 at each age was ob-

FROM THE RECORDS OF INSURED LIVES. 67

tained, since these extreme ages are each included in two groups.
The values of l[x-2j4]+}^, hx-2yi]+i}4, hx-2H]+il4f ^^c, were next
calculated, being based upon the 50,000 assumed for the value
of Z[i_2j4]. Similar values for each age at entry were calculated
and a combination of the values for the same durations for ages
at issue {x - 2^), (x — U), (x - ^), (x + h)> (^ + 1^) and
(x 4- 2^) were taken as the values for the average age x at entry,
the value of l[x] being 500,000. The total deaths at the end of
^, If, 2f, etc., years, were then found and by an interpolation
formula the total deaths at the end of 1, 2, 3, etc., years were
calculated. Then by taking the first differences of the results the
numbers dying in each year of insurance could be obtained.
Having obtained in this way the necessary data for quinquennial
ages at entry, graphic graduation was used to adjust (a) the
rates of mortality for the first insurance year, (6) the ratio of
the rate of mortality of the second insurance year to that of the
first, and (c) the corresponding ratio between the third and
second insurance years. The values of q[x]+n for n = 0, n = 1
and n = 2 being thus available, and for n > 4 being taken as
those of Qx+n by the H^^^) Table, values for n = 3 and n = 4
were supplied by interpolation. Interpolation was used also
to supply the values for other than quinquennial entry ages.
In some cases, for values of n less than 5, H^^^) values of qx+n
were substituted for those found as above described because
Qx+n by the H^^^) table was less.

A corresponding method would have to be followed in order to
obtain select tables from an experience based upon life years.

However, when it is possible to obtain suflScient data select
tables should always be constructed by a policy year method and
not by an indirect method such as that employed by Dr. Sprague.

Construction of Mortality Tables from Limited Data: It not
infrequently happens that the data upon which a mortality
table is based are not sufficient to give reliable, if indeed any,
results at the young and old ages. Where the data are graduated
by assuming some law of mortality, as described in Actuarial
Studies No. 4, the values for the young and old ages may be
supplied by assuming that the law holds throughout the table.

When a table is not so graduated it may be necessary to sub-
stitute for certain portions of the table which is being prepared,
the data from some standard table which are believed to approxi-
mate to the mortality of the class of lives on which the new table

68 CONSTRUCTION OF MORTALITY TABLES

is based. The junction of the data of the standard table with
that of the new table may be smoothed off by graduation or
interpolation.

Where the data are very limited throughout, the actual deaths
may be compared with the expected deaths according to some
standard table, grouping a sufficient number of ages to secure
enough data to give a general idea of the ratio of the actual
mortality to that shown by the standard table at various ages.
The ratios for intermediate ages may then be found by interpola-
tion and the new table formed by applying the adjusted ratios
to the data of the standard table.

Homogeneity of Data: In constructing a mortality table our
purpose is usually not only to ifind out what rates of mortality
have been experienced among the lives investigated but also
to have a table which can be applied to estimate future events.
The latter purpose is the chief one in practically all cases. If
we are to be able to apply a table with any confidence we must
know the character of the lives that contributed the experience
and any special conditions to which their mortality was subject.
It therefore becomes obvious that our material must be suffi-
ciently homogeneous so that the results can be said to apply to a
reasonably stable group of lives.

A little consideration, however, will show that when we speak
of "homogeneous data," we are using a relative term. It is
impossible to imagine a large group of lives which is not made up
of many subgroups each of which is subject to slightly different
mortality rates. For example in insurance experience, those
lives which are insured under plans calling for the lowest premium
rates will, generally speaking, show the highest rates of mortality.
Where those insuring have a free choice as to whether their
insurance shall be upon the participating or non-participating
plan, it will generally be found that those selecting non-participat-
ing policies will show the higher mortality, though there is not
the same reason for supposing that a company issuing only non-
participating policies will experience a higher mortality than
another company granting insurance only upon the participating
plan. The relative mortality of male and female lives differs
at various ages. The residents of different sections of the
United States are subject to different rates of mortality. Even
if we disregard outside influences we shall find degrees of difference
among lives insured in one company at standard rates due to

FROM THE RECORDS OF INSURED LIVES. 69

physical or occupational causes. Then again mortality rates
will be influenced by such external causes as agency methods,
or, over different periods of time, the advances made in medical
science and sanitation and in the science of selection.

Mr. G. F. Hardy* has given the following statement of the
problem and its proper solution.

"The Actuary constructs tables not merely to show what has happened in
the past, but to enable him to forecast the future, and as he requires these
tables as a basis for financial operations, considerations are introduced which
do not arise in the treatment of purely statistical tables. Whatever class
of events the Actuary may have to deal with, will be subject to change with
the lapse of time. That portion of the class he has been able to observe hes
necessarily in the past; the conclusions he has derived from their study he
proposes to extend to the future. He must therefore consider how far the
observed characters of the class are changing or permanent, and must en-
deavour to distinguish between changes representing permanent tendencies
and those due merely to temporary fluctuations. In the selection of data
suitable for his purpose the Actuary will aim on the one hand at a sufficiently
broad basis both in space and time to eliminate the effects of local and tem-
porary fluctuations, and on the other hand he will aim at obtaining as far as
possible a homogeneous group of data. These two aims are more or less in
conflict, and he will lean to the one side or the other, according to the object
he has in view. Where, for example, that object is to produce a table that may
be adopted as a general standard by various institutions, often differing con-
siderably as to their individual experience, he must aim at a correspondingly
broad foimdation. In these circumstances it will not generally be possible
to obtain a really homogeneous experience. If it is a question of the mortaUty
of assured lives, for instance, this will be found to be affected by endless in-
dividual variations, age, sex, duration of assurance, occupation, civil condi-
tion, class of assurance, character of the insuring office, etc., etc., and from
such material approximately homogeneous data could only be obtained by
cutting up the experience into comparatively small groups and thus sacrificing
all generahty. This can be avoided in practice by first excluding all extreme
variations. The sexes will be separately treated, lives so impaired as to
prospects of longevity by personal health, family history, occupation, or
residence in unhealthy districts as to be "rated up" will be excluded, as also
classes of assurance that may be supposed subject to rates of mortality dif-
fering from the average. When the data has thus been trimmed of the ex-
treme variations, a body of experience will generally remain not greatly
shrunken from its original dimensions and in which the discontinuous varia-
tions are sufficiently numerous and individually unimportant to render the
data for practical purposes homogeneous. The rates of mortality, or with-
drawal, can then be treated as functions of the two remaining variables of
importance, the age and the time elapsed from date of entry; or as functions
of the age only from the point at which the factor of duration may be found
to be unimportant."

* "The Theory of the Construction of Tables of MortaUty, etc.," p. 16.

70 CONSTRUCTION OF MORTALITY TABLES

Spurious Selection: After a reasonably homogeneous set of
lives have been taken to form select and ultimate tables there is
still some question whether the effect of the selection of risks
can be measured precisely by the difference between the ultimate
rates and the select rates for the same attained age but for dura-
tions within the select period ; nor can we be altogether sure that
the effect of selection is felt for the exact period shown by the
tables. In the first place it will be noted that the practical
necessities of graduation may lead to the choice of a uniform
period of selection at all ages at entry contrary to the evidence
of the unadjusted figures, and may also result in an appreciable
change in the select rates for the later part of the period at some
ages in order to form a smooth junction with the ultimate rates.

Mr. W. P. Elderton introduced the term "Spurious Selection"
(J. /. A., Vol. XL, 221) to describe "the selection indicated by a
difference in q which has arisen entirely from statistical proc-
esses." He examined the case where a table is formed by
amalgamating two classes of data to which he applied the symbols
Eix]+t and E'ix]+t, the prime indicating the data subject to the
heavier mortality rate. Using q^+t and q'x+t to indicate the
respective ultimate rates of mortality to which the classes are
subject, the rate of mortality that would be found for the
{t + l)th year from the amalgamated data would be

Eix]+tqx+t + E ix]+tq x+t
E[x]+t + E'ix]+t

Similarly the rate for the same attained age arising from the
next year of duration is

Eix-i]+t+iqx+t -\- E ix-i]+t+iq x+t
Eix-i]+t+i + E ix-\]+t+i

The rates qx+t and q'x+t are used in both cases since each class of
data is assumed to have passed beyond its period of selection.
It is seen, however, that if q and q' are unequal the resulting
mortality rates for the amalgamated table will be unequal though
they apply to the same attained age. If the latter expression
is the greater then selection will appear to be still effective
whereas we know it has really ceased, but if the former is the
greater we shall get the opposite effect to that of selection, or
what may be called a tendency to conceal the effects of selection.

FROM THE RECORDS OF INSURED LIVES. 71

The analysis may be carried further, for

Eix]+tq + E\x]+tq' > E[x-i]+i+iq + E'[x-iu-t+iq'
Eix]+t + E'ix]+t "^ E[x-i]+t+i + E\x.i]+t+i
as
^lx-i]-i-t+iE\x]+tq' 4- E[x]+tE\x-i]i-t+iq

= E\x..i]+t+iE[x]+tq' -\- Eix-i]+t+iE\x]+iq
or as

Eix-i]+i+iE'ix]+t{q' — q) ^ E[x]+tE'ix-i]+t+i(q' — q)
or as

E[x] + t "^ ^[x-l] + (+l ■

In other words we shall get the appearance of selection that
does not exist if the right-hand member of the last equation is the
greater, i.e., if the proportion of data subject to the heavier
mortality is increasing with duration for the same attained age.
We may come to this conclusion also from general reasoning,
for in such a case the ultimate data will contain an undue amount
of the heavier mortality data while the very early years of dura-
tion will be largely made up of the lower mortality data, so that
in order to have the mortality rates pass from the low initial
select rates to the heavy ultimate rates will require us to pass
through an increased number of years of duration. Thus the
apparent effect of selection is increased both by increasing the
differences between the select and ultimate rates and also by
increasing the period of selection.

If, on the other hand, the left-hand member of this equation
is the greater, we obtain the contrary effect, our initial select
rates being increased by the heavier mortality and the ultimate
rates being lowered by the larger proportion of lower mortality,
thus tending to offset the true effects of selection.

The former condition would seem to be the more likely one in a
life insurance experience because of the fact that, generally,
there has been evidence that mortality has decreased with ad-
vancing calendar year of entry. The longer the duration the
larger would be the proportion of data arising from early issue,
if the investigation is terminated for all policies on a certain
date or in a certain year.

Mr. P. C. H. Papps investigated another special case {T. A. S.
A., Vol. XIII, 211) in which spurious selection may be due to a
progressive improvement in mortality with advancing calendar

72 CONSTRUCTION OF MORTALITY TABLES.

years of exposure affecting all the data irrespective of year of
issue. Such an effect might be due to improvement in the general
health of the population. He assumed further that the data
included only policies issued during the period of investigation.
He has shown that, for an equal volume of issues each year, the
result would be to produce too low an effect of selection.

This result could also be reached by general reasoning for the
mortality rate for the longest year of duration would be deter-
mined from exposures in the last year of the investigation, while
the rate for the first policy year would come from exposures
over the entire period, with intermediate durations passing from
one extreme to the other. Consequently the mortality rates
for the early years of duration would be relatively high com-
pared with those of the later, or ultimate years.

As pointed out by Mr. F. H. Johnston, in discussing Mr.
Papps's paper, this effect vanishes when a table is constructed
from data that includes the experience of issues prior to the
period of investigation, assuming that the distribution of the
exposed to risk for any one policy year was chronologically the
same as for any other policy year — which is fairly close to the
fact. In such a case the proportions of the exposed to risk at
duration t arising in the first, second, third, etc., years of the
investigation is the same as the proportions for other durations.*
Then the mortality rate for each year of duration would be
equal to a constant multiplied by the rate for that duration for
any specified year of the investigation. This would also be
true if the mortality rates changed in any manner from year to
year of the investigation.

The question whether spurious selection exists is a complex one
in any case, and its importance in any table constructed from
carefully selected data in the usual way is probably small. It
does emphasize, however, the need for obtaining homogeneous
data if we are to place much reliance on the effect of selection
displayed by the resulting tables.

* This woxild not be true in general for the longest durations, but this would
have little e£fect on the ultimate rates in most cases.

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Full text of "Construction of mortality tables from the records of insured lives;"

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